Calculus 2 Lecture 9.1: Convergence and Divergence of Sequences

All right we can to start something completely new uh we're going to start talking about chapter 9 in chapter 9 we talk about sequences and then Series this is a complete s side step from all the integration techniques that you've had up to this point we will be using them but it's not just integral after integral after integral Ed that's not what we're talking about here we're going to be talking about these two Ideas called sequences and Series in this chapter and in the first section we're going to limit it just to sequences we'll talk

about what a sequence is I'll show you how to get terms from a sequence get a sequence uh in sequence notation from a series of term from a sequence of terms um I'll also show you how to tell when a sequence will converge or diverge those two ideas are going to be very important for us in this chapter most of what We're going to be doing is testing for convergence or Divergence whether a sequence or series goes to a uh one number or whether it doesn't that's in the basic sense of things so in section

9.1 we are going to get a very indepth discussion of what sequences are all about again I'm going to try to teach this teach this to you because if you again if you're like me you didn't get a Lot of exposure to this before right now uh which means that you might have seen a sequence before but you really don't know a whole lot about them we're going to learn a whole lot about them right now first thing what a sequence is so a sequence sequence typically denoted like this a subn it's a set of

terms and the set of terms is based on well whatever this sequence is given to You as it's typically going to be first term second term third term and we denote those as a sub 1 a sub 2 a sub 3 all the way until we get a subn wherever we start n is always going to be a positive integer for us we we can't have the neg a negative term here so we say hey we have the first term it's it's a sub one sometimes you can do like the zeroth term a sub zero

but typically we have a sub 1 a sub 2 a sub 3 a sub 4 so n is going to be our positive integer Also these sequences can be denoted like this you don't see this very often at least not in our class it can be given to you like this where we say your 's going to start at a certain spot and it's going to end at a certain spot or it doesn't end we can denote a sequence like that so this says hey here's what you're going to be working with this is going

to be some sort of a notation like n + 1 over n or n^2 over n factorial it's going to give You it's not just going to be a subn okay it's going to give you something there to work with then it says Hey I want you to start your n at three or at five or at one and I want you to end at seven or at 12 or I don't want you to end I want you to go forever and ever and ever and we'll talk about that does that make sense if this

is not shown so if you don't have that then n starts at one so if this isn't here if not Shown n starts at one now what I'd like to do is go through a couple of examples of just what sequences look like just so we can get the habit of listing out terms seeing what these things are actually doing would you guys like that be helpful because right now are you a little bit confused on what sequences are a littleit okay some of you probably have never seen sequences before so here we go here's

an example what we're going to Do is we're just going to start listing out some terms for the following sequences so here's how a sequence could look I want to make it look a little like what the heck just so you see that it's not that big of a deal list up terms they're very easy to list up terms okay that right there is something that's in sequence notation we have something that's do you guys see how this is in terms of n it says hey here's this the sequence in terms of n it says

Well where's the n g to start for us one why is it going to start at One automatically it's not noted if I'm not giving you anything over here this thing's going to start at one so here's how we find out the terms of our sequence it's not hard all it is is you start wherever I tell you to start if I don't tell you to starting where you start at one that would mean that our first term a sub one would happen when I plug in one for ourn so for Instance go okay well

uh what happens when when n is one that would be our first term what's going to happen when n is 2 what's going to happen when n is three or n is 4 or n is 5 by the way is this ever going to end for us so we're just going to list out the first five terms just to get a feel for it so if I want when n is one what I'm going to get out of here we're going to call this a sub one it's going to be My very first term it's

going say the sequence of a so that's this thing when I've plugged in one or basically the first term of our sequence Jo P be okay with the idea here it's not hard to find out you just got to plug in one so let's plug in one where do we plug the one into again okay so um let's work through it what's n + 1 when n is 1 two how about1 to the second what's that one okay this would be 2 to the 1 how much is that would be 1 * 2 so we're

going to Have two over what one perfect yeah we're going to get one out of that you guys okay with that so far I'm going to erase that and put it one but that's where it's coming from it's just plugging in one and doing a little simplification so here's how we' say this the first term of our sequence or a sub one is one how we got it was we plugged in one for our n uh you don't Have to show all this stuff basically we just show first term second term third term I'm just

kind of denoting this so you understand where what all this stuff means okay a sub1 means the first term of our sequence a Nal 1 means well that's how you find the first term you plug in the one does that make sense let's plug in the two so this would be our a sub 2 or our second term of our sequence how you find the second term of our sequence is if you start at one you Plugged in one if you start if you want to find the next term you're going to plug in what

two for what and plug it and plug in the two so all right plug in the two if I do 2 + 1 notice I'm doing two here 2 + 1 is 1 to the third is okay so we know negative this is going to be negative basically that's all it's saying 2 to the 2 is so this will be negative 4 over we're pluging in two so the first term we got one the Next term we got 4/3 here's my next question to you I want to see if you really paying attention here right

first term was positive second term was negative third term what do you think that's going to be how about the next one how about the next one how about the next one is it going to do that forever yes because this right here that negative one that doesn't change value but it will change sign and says hey plug in a number and add one to it it's Either going to be positive or it's going to be negative because we're going to get an even number or an odd number is an exponent does that make sense

to you this right here gives us that alternating type of a sequence positive negative positive negative or negative positive negative positive my point is do you guys see that this is going to do that forever yeah okay so next term is going to be positive for sure let's check it uh how Would I find a sub3 what would I plug in you just look at the pattern and go through say what oh never mind you got to do a first yeah but I'm saying I notic there's a pattern there it's going to be positive negative

positive negative posi not only that but there's going to be a pattern for sure all these most of these things do have a pattern to them for us uh so plug in the three 3 + 1 four Four 1 4 is positive it's positive I don't care if it's one or not it's going to be positive one or netive just telling it's positive because I'm multiplying by something right this is going to be eight so I'm going to have positive 8 over you guys okay with that one yeah let's do the fourth term why don't

you do the fourth and the fifth terms for me okay try the fourth and the fifth terms for Let's do that uh when you plugg in the fourth pluged in for the fourth term so basically when I'm looking for a sub4 I plug in four to my N I know if I plug in four 4 + 1 is 5 that's an odd number therefore I'm going to get a negative 1 5th is going to be a negative uh term here so this should be 16 over 5 Nega 16 over 5 did you get the same

thing yes and lastly for a sub5 just plug in five I know my sign's going to change go from negative to positive Because I'm taking hey look at that I'm taking 5 + 1 if I plug in an odd number it's going to add one to give me an even number any negative to an even number is going to become a positive so that's positive 2 5th is2 32 over 6 32 over 6 is 16 over 3 POS 163 okay really do need to show FC feel okay with that again is this sequence ever going

to end no so we just found The first five terms but notice that it doesn't end forever you can do this forever andever it's always going to alternate positive negative positive have I explained well enough why it's alternating from positive to negative and then negative to positive and back and forth and back and forth have I explained that well enough for you here's my next question and let's look at a different a different sequence is this sequence going to Alternate positive negative positive negative positive or negative negative positive is it going to do one of

those things what tells you immediately that this is going to be an alternating sign on these terms right there that's it when you see this that's going to be alternating for sure because we're going to start off either with a positive or with a negative but the next one's going to be a different number right next one's Going to take that negative one to a different consecutive number so if we start with an even the next one's going to be odd the next one's going to be even the next one's going be odd and then

even odd and negative one to an even versus an odd is a difference between positive and negative should feel okay with with that one so we're going to have this idea again now next tell me something interesting that happens here that we didn't happened on Our first sequence do you see why on this particular sequence I'm starting at two yeah yeah I mean if I started at one I had to get an i and well we're not talking about that in this class we're talking about real numbers so this would be undefined for the real

number system does that make sense to you so we got to start at two for this one let's go ahead and I want you to figure out the first five terms of your sequence go for it First five terms you don't have to do anals 1 and equals you don't have to show me that you don't have to even show a sub 1 a sub 2 a I'm probably going to do the a sub 1 2 3 4 and 5 just so you see what what they are but you don't have to do that all

I'm asking you to do is when you start here well we would start at a sub one we start at a sub two we'd start a sub 2 then we do a sub 3 a sub 4 a sub 5 and a sub 6 does that make sense yeah It will be the first term but it's a sub2 because that's our ter plug in bless you thank so I want the a sub two a sub 3 a sub4 a sub 5 and a sub six terms basically we're plug in in 2 3 4 5 and six

it's not going to end we're going to stop but this thing doesn't end it goes to Infinity we just have to start with a sub 2 because it says well wait a minute you're starting at Nal 2 because we listen to our terms a subn You got to you got to signify that we're not going to put look at we're not going to put a sub one are we plugging in one no then you wouldn't put that you plug in two you got to have a sub2 that's where we're starting this particular sequence when you

plugged in uh the two for your n what did you get zero Z 2 - 2 is z doesn't matter what the rest of it is it's going to be zero you okay with the zero yeah when we plug in the three well if I plug in Three a sub3 means take your n and make it a three this is why our notation has to be correct because it's telling you what you just plugged in what you're supposed to plug in take that three uh what's 3 - 2 1 one okay now this is-1 to

the N just to the 3 we don't have to add or subtract anything to it what's -1 to the 3 power positive or negative so this is going to be negative < TK 1 or 1 does that make sense to you did you Make it that far yes good let's plug in the four if I want to find a sub 4 I'm going to plug in four 4 - 2 is 2 now now I'm going to do the fourth power remember this for four four goes here four goes here- 1 to the 4th is a

positive 4 - 2 isun of two I don't know sleep I don't know what that is I'm not going to do 1.41 or anything I'm leave it square of two I want to be exact with this stuff quick head not if you're with With that one yes okay good two more a sub5 says all right take that five plug it in for your n so a subn a sub5 hey five goes here five goes here tell me am I going to get positive or negative netive yeah should be a negative 5 - 2 is 3

three I don't know leave it of three next up a sub 6 6 is going to go here six is going to go here so1 the 6 power positive or negative for sure 6 - 2 is 4 Positive two it doesn't end it goes forever now here's what's interesting you will find patterns in these things first pattern we see is that the signs are changing positive negative positive negative positive negative second one this is just the square root of 0o the square root of one the square of two the square Ro of three the square

root of four four next one will be negative square Ro T of 5 next one will be positive of 6 next one be negative and So on and so forth and so on and so forth forever and ever and ever and ever does that make sense to you they have a pattern to it of course it's a sequence it's based on one sequence here it's going to have to have some sort of a um sequential order to it of course we'd simplify them which takes away that look of a pattern doesn't it they don't look

like it has a pattern so sometimes what that means is that if I give you this and I Want this from you so right now I've given you the sequence I wanted the terms what if I gave you the terms and asked you for the sequence could you do it yes but it's uh it's a little bit harder cuz you're not given this to just plug numbers into now I want you to go from here and give me this sort of thing and that can get a little bit tricky so I'm going to walk you

through a couple examples on how to do that but I want to make sure I've explained this well Enough you understand what a sequence actually is so dep if I have feel okay with that questions what with the negative I'm not sure what the me on the top of the like I mean negative sorry uh infinity sign means it doesn't end oh okay so if I said hey you know what I just want uh up into n six we would have stopped right here not that make sense yeah okay let's keep on going so like

I said we can go backwards Too where if I give you a list of terms we should be able to find the sequence notation for that by the way it is appropriate I didn't do this but it is appropriate to when you're writing out a sequence to write these symbols as well to show that this is the sequence that we're getting here so we really should have that I didn't do it but we should have it say that is my sequence they're the same Thing you okay with that one okay so as I was saying

if I give you a sequence notation like this where you just plug in in this is pretty easy it's pretty straightforward is it straightforward to you yeah plug in the ends not a big deal but because these have a definite pattern to them that means that from our sequences given to his terby term rotation we should be able to get this thing back again and that's what I want to do with a couple Examples here to give you the idea now naturally all these are going to be different some are going to be way harder

than other ones all right some of you are going to be much better at others of you that makes sense than others because you're good at seeing patterns um I I struggled with this when I was taking this class man I can't see the pattern took me a lot of work uh it's hard sometimes to see these patterns I'm going to give you the way That I went through this um that helped me out a lot the first first thing I did was I listed out all the terms that were given to me so I

would do like look like a nine supposed to be a I would list out what a sub 1 is a sub 2 a sub 3 a sub4 until they stop giving me terms can you tell me what is my a sub one everybody what's my a sub one good what's my a sub 2 very good and then 4 < TK 3 and five square sorry 5 over Two the next thing I did was I tried to write them in all the same form what what I mean by the same form is look for a pattern

that most of them have and some of them don't have and try to make them fit this this format what what I mean by that is this um most of our denominators have a square root of around them at least some of them do so what I'm going to try to do here is I'm going to try to make all the denominators have a square root around Them does that make sense that'll give us an idea of a pattern so if I say hey you know what two wait a minute instead of two making yeah

this is 2 over one moreover I can do two over root of one notice how I changed that term the look of that term not the not the value of it it's still the same thing right 2un one is still two but now it it more fits my pattern does that make sense to you this is how I did finding a Pattern then I would do well this is pretty good this is 3 overun 2 no problem this is 4 over < tk3 that's not an issue this one's a little different though 5 over two

I don't like thatare aha let's make that have a square root since that has a square root that has a square root probably this one's going to have a square root but but squ of two is not two TK of 4 is two show F feel okay with that one so your first idea my first idea was Let's write out all the the terms boom boom boom boom let's try to make them have all the same pattern so if I seen some square roots here I know that my sequence probably has it has to have

a square root somewhere in there does that make sense it's got to have one somewhere in there so if I have a square root for my denominators let's write our denominators having a square root 2 over 1 hey call of one two let's call itun of 4 now it's still of one of two of 3 of 4 So you okay with that one okay next idea try look for a pattern that is based on some sort of an N so the idea is ultimately try to relate your terms to some n now when I say

N I don't mean some arbitrary number what I mean is look up here at the board here real quick what's your n Rel relate this to one relate this pattern to one what's your n relate this pattern to two what's your n relate to pattern three does that make sense to You so here's how I'm do what I mean by that I mean that well the first part of this for me at least is really easy because what's nice is that if I see one number that always matches up look at this look at the

board right now one one hey one one 2 two hey 2 two 3 three that's great 4 four I don't have to change that that's brilliant so I know the of 1 and < TK of two and < TK of 3 and < TK of 4 means that I'm going to have a square Ro T so We want this right this is ultimately what we want to get to this is it if I have hey 1 one 2 2 3 3 4 4 then I'm going to have n n that's exactly right this is the

idea we want to go from the specific to the general we're going from the inter the the exact terms to how can I do this every time and you can always double check your work okay always double check your work if I plug in one do I get that yes if I plug in two do I get that a sub two two hey if I Plug in three I get this if I plug in four I get this so what I'm doing here is I'm trying to make all these things have the same format that

way I can see a really easy pattern to it hopefully it relates easily to my ends some of them don't some of them are hard to see some are really easy to see this one I would say is easy one to see there's a one and a one a two and a two a three and a three and a four and a four you guys got it yeah now we got to do The same thing with the numerators here I got a two but I start with a one here I got a three but I

start with a two here I got a four but I start with three here I got a five but I start with a four what's a relationship so this is really two is really 1 + 1 relate it to your end three is really look at this three is really 2 + 1 four is really 3 + 1 5 is really 4 + 1 a subn for the numerator is really n + 1 relate it to your n does that make sense to you yeah so your idea here and it's not as easy as probably

I made it look okay this one was an easy example uh the idea is try to make them all the same format if you got square roots put square roots everywhere if you can do that okay try to relate it back to whatever your your index is so a sub one related it back to one related it back to two Rel it back to three to four and do that with both the numerator Denominator whatever you have you're always trying to relate it back to the given end for that specific term if you can do

that if you can write everything in terms of this index everything in terms of this index then it's really easy to write your your sequence in sequence notation you okay cool well look at what's happening this is 1 2 2 3 3 4 4 so this is n n n and n that means an N has to go here one one the one's not oh I forgot the One the one's not changing okay well the one's not changing we're just adding one to it this is NN n n that means an N has to go

in the side of the square root that means that this sequence is represented by this sequence this is the sequence n + 1 < TK n the way that we do that we say hey you know what if we continue this pattern forever folks if we continue this Pattern forever every term is going to be given by this this is the formula for every single term that is the formula for our sequence hands if I if you understand that cool you want to try another one yeah okay completely different topic from our integrals isn't it

completely different right now we are going to see how integrals do take place in series not so much in sequences it's going to be more Of a limit idea but we are going to see them eventually um but it's it's like AEP it's a different type of thinking process to me it's interesting it was hard the first time I saw it I really didn't grasp it all that well the first time I did this oh my gosh it's brand new so if you're strugging with that just know that you can you can get it uh

but you have to kind of switch your mindset a little bit to It's thinking outside the box some of the times let's Do one more okay so there's our sequence the first thing that I would consider doing is writing out I would blindly look for a pattern sometimes you can do that and sometimes oh my gosh that's so obvious to me that's cool if you're that type of person man more power too that's awesome uh for me I'm not all right I have to go through this process so for me I would write out a

sub one I would write out a sub 2 a sub Three a sub four how many I give you here five I'd write out all of them I'd go through the process so I start with just the terms themselves well I try to make them all the same format now the same format for me means that I've got fractions everywhere that's the idea so I got fractions so the only thing I do is I change this into a fraction after that there's no square roots I don't see any do you see Any exponents I don't

see exponents s seeing square root so as far as the format change that's all we can do the next thing is we got to try to relate this back to our n we have to do that so uh first thing first thing's kind of nice um I see something as far as my pattern tell me one thing that you see about the pattern here one top okay there's a one there's a one always on my numerator well that's going to make it kind of Nice right sign change oh sign change that's right so sign change

so here's here's the one thing I know right now I know that my sequence is going to have terms where there's going to be a one on the numerator that's what I know you guys know that too right okay well then now that's going to change right it's sometimes be negative one sometimes going to change to be a positive one so I know that here it's Negative but here it's positive here it's negative but here it's positive here it's negative but here it's positive it's going to change uh the whole way through this here's how

we deal with that alternating idea the alternating signs idea remember the first two Ser sequences I gave you it was negative 1 to a power you're going to either do 1 to the N or1 to the n + one depending on whether you start with a positive or Negative so let's think of about it do we start with a positive or A negative remember that we're always going back to whatever our n is here our n is one so what I want to know is that which one which one is going to give us a

positive sorry a negative for the first term and a positive for the second term if I plugged in one here if I plugged in one here I'd get 1 + 1 is 2 1^2 is positive that would not be the correct case here If we take away that plus one basically we get this one we plug in n = 1 1 to the first is still Nega 1 that means that's going to give us this one if I plug in a two here it would give me a positive okay if I plug in a three

here it give me a negative okay four here positive five here negative I'm always relating it back to my n whatever the index is does that make sense to me you sure that's another piece of our puzzle so we're going to do something real Quick instead of having this one up here would you be okay if I just did1 to the end yes that's either going to give us a positive one or negative one I know the numerator is going to be one or negative one but this this this gives it to us let's double

check before we go any further okay double check our numerators relate it back to our index if I plug in one do I get negative one yes yes so I can move those up here even if you want if that's helpful to you do it if I Plug in one I get 1 if I plug in two do I get positive one I plug in three do I get negative 1 four do I get positive one five do I get eight you know what it's going to work then that's our we already done our numerator

now it's just the denominator so let's work on the denominator take each piece of this and relate each piece back to your n your index now right now that looks like the n and that looks like the N right does This one look like the N so I can't just put Over N so I put over n the first two would be right the rest would be wrong I've got to have something that's right for all of them so let's try to relate this sometimes it's nice to not look at the first one doesn't really

give you a whole lot of information here let's look at maybe um look at maybe the second one on down do you see a pattern out of That what do you see specific how do you get from 2 to 6 how do you get from 6 to 24 how do you get from 24 to 120 how do you get let's try easy one okay don't add to it because this is not addition here how do you get from two to six multiply by how do you get from 6 to 24 how do you get from

24 to 120 five so what this is doing it's Taking the previous number it's multiplying it by your n now hang on a second so this would take the previous the pre the previous number doesn't exist it just multiplies it by one one takes the previous number look at please look please follow the pattern here takes a previous number and multiplies it by remember you're always ring back to your n do you got it always this is where it's going to get interesting please listen carefully the Previous number is 1 * 2 correct it multiplies

it by n the previous number is 1 * 2 * 3 correct it multiplies it by n the previous number is 1 * 2 * 3 * 4 * it multiplies it by factorial ah now we're getting somewhere can we rep repr this better yes this says take n multiply by every number before that that's what this is doing it says this number is 2 * this one this one's 3 * this one this one's four times well all the previous ones Because that's how we're getting those ones we're getting them by multiplying the ends

from every previous term does that make sense to you this is how you find the pattern though it's not that easy to look at it most people unless you're really good at factorials would see one 2 3 4 and 5 factorial I did because I've done the problem before uh but probably not if you're first going through it you got to really look at that pattern does that make sense to you If we do look at the pattern and go hey look at that this is 5 factorial four factorial this is 1 over 1 factorial

this is 1 / 2 factorial this is -1 over 3 factorial this is 1 over 4 factorial this is1 over five factorial by the way if you don't know what factorial means this is not five okay this is a five factorial I know it's exciting almost good to talk about radicals but uh but it's not this is five factorial means 5 * 4 * 3 * it's uh A product of the number that I give you and every previous consecutive integer until you get down to one so when we relate this back to our n

we've already taken care of our numerator it's great it's alternating one not a problem denominator starts with N Factor that's what it is therefore this sequence every term of this sequence a subn this is what you're looking for you're looking for a subn every term is Given by this and you should definitely double check your work before you go any further make sure when you plug in these numbers you get the same thing that'd be really silly to do this and have wrong okay plug in one go uh yes that gives me one plug in

two and that gives you 1 12 plug in three that gives me6 you should be doing that okay after you've done it You' verified it this is what goes inside your sequence notation -1 to the N Over N Factorial and we just solve have I explained well enough how to find your sequences from your terms that's way harder than finding your terms from your sequence okay that's pretty easy just plug numbers in very straightforward that Way's kind of hard now I want to give you a couple other ideas before I I leave you today the

first one is a kind of an interesting idea sometimes we can Define sequences by Giving one or two or three or four or finite number of fixed terms and then defining our sequence around those terms for instance if that's like what in the world did you say in English uh here's what I said in English give me a couple numbers uh like two and four let's pick two and four so don't give me nothing I'll do it on my own thanks a lot guys so I say a sub 1 is 2 2 a sub 2

is 4 I'm not looking for a pattern right Now I'm not thinking that a sub 3 is 8 or 6 okay I'm not thinking that here's what I'm going to Define for the rest of them I say based on these two terms your next term a sub n + 1 your next term is given by 2 * a subn minus a sub nus1 you go what in the world is this this is called a recursive sequence it's called recursive because it relates back to the previous terms a recursive idea is look back at what you

had and build From there it's based on what happened before it so this is called a recursive sequence what I the only thing I'd like to do right now is U come up with the next few terms so let's see how this actually works it's not hard it looks weird okay it really does look weird but I want you to think about what's going on here if you're wondering wow what is this a sub n + one that just means your next term so for instance for instance what terms Do I know right now a

sub 1 a sub2 that's what I know so what term would I be looking for so watch carefully a sub3 would therefore be equal to check out this the way the recursive formula works it says if you want a sub3 you're going to consider a sub3 to be a sub oh here's that would be 2 + 1 so then a subn would be just the one before it a sub 2 minus oh look at this a sub nus 1 would be a sub 1 did you guys get that Idea where those things came from from

yeah are you sure yes if this is my formula this is my recursive sequence it says if this is a three if this is a three that's one more than two this is one more than one so I'm looking these terms a sub 1 a sub 2 oh that's a sub3 do you see how it's based on the previous terms there it's based on what I give you if I change these two numbers this whole sequence Changes so P feel okay with that idea let's find it so a sub3 therefore would be 2 * what's

a sub 2 four minus what's a sub 1 which means that a sub3 is oh look at that it didn't turn out to be six how about that interesting let's find two more I'll do one more with you you do one your own a sub4 it might be helpful for you to right now write out what I did here First this will give you a much better idea what's going on right now so watch this please I know that a subn +1 says hey take two * the pre that's that's all this means it just

means the previous term this is a sub3 this is a sub 2 this is be a sub 1 if this is a sub 4 this is a sub 3 and this is a sub it's just the previous two terms so this would be 2 * a sub 3 - a sub 2 if that makes sense to you you got it Do you got it are you sure you got it for for that last one we did did we have done n * a sub nus1 like n * the previous answer or would it not work

in this is literally what that means did you catch that yeah I was just wondering if we that not this is this is a factorial if you didn't think about that start with the number take one away from it take two away from it take three Away from it multiply them all together till you get down to one we're actually going to use this in a minute so are there different ways you can Define your sequence yeah probably um but we're going to try to do the nicest way that makes sense did that answer your

question at all I'm talking about doing the a sub just saying to Define this one recursively say like a sub one is1 um I don't know that you'd want to I think that I'd probably want to stick With this way and not if I don't want have to Define recurly I'm probably not going to because I want to formula to be able to find the tell tell you what we want to do you want to be able to find the 10th term without having to find the first nine okay um you can't do that with

recursive sequences most of the time because you'd have to know the previous couple terms right find the 10th term you'd have to know here number nine and Number eight does that make sense so if you can do it without recursive I would say do it without recursive now let's finish this off uh real quick do you all understand where the A sub3 and A sub2 came from yes that's all this says is the previous two terms so in our case we'd have two hey what's a sub3 notice how you had to know that before you

did this a sub 2 so a sub4 = 8 how about a sub 5 a sub 5 would be 2 * a sub 4- A sub3 it's just the previous two terms this is the idea of a recursive sequence now please listen carefully is it always a previous two no it can be different it can be the previous term and then three before that you just have to have it defined okay so a sub5 = 2 what's a sub4 8 minus a sub3 6 a sub5 therefore is so we know that this sequence goes like

this it goes 2 4 6 8 10 and it never stops it's just the even numbers that's Exactly what that is so if F feel okay with this idea all right good deal so continuing on with our section we're going to talk today about the limit of a sequence and what that infers as far as convergence and Divergence so what we need to know is that if we have a SE sequence a subn we can find the limit of the sequence and it's going to be written as the limit of a subn we'll do some

examples on what this means in just a little bit as n Approaches Infinity you see these sequenc sequences like this are implied to start oh we learn this last time implied to start where if I don't tell you anything it starts at and it goes to so what we're trying to find out is what's happening to our sequence as we get bigger and bigger and bigger is the sequence getting bigger and bigger and bigger forever or smaller and smaller and smaller forever more negative or is it tending to be towards one value and That's what

this limit idea is going to tell us whether our sequence is going to one number hence it will be convergent if a limit exists or if it's going to infinity or negative infinity or whether the limit doesn't exist in that case it would be diver diverent that's exactly right so to find out what's happening to our sequence we take a limit for the limit of our sequence we're going to have a couple results where I put that if the limit exists so equals an Actual number then we say that that limit then the sequence is

convergent so if the limit of our sequence as n approaches Infinity exists then the sequence is convergent if the limit does not exist so it does not exist or is infinity so if not I'll say it this way so if the limit does not exist to some number if it's infinity or negative Infinity so if not then the sequence is Divergent Now a couple properties that I'm going to go through super fast these are old properties for limits I just want to refresh your memory on what these things first is this is this okay is

this explained well enough for you to understand that trying to figure out what's happening to our sequence right so we're going to take a limit as the terms as the terms go towards Infinity as end approaches Infinity so way out there we're going to see if it Approaches a number or not if it apprach is a number hey it's convergent if not it's Divergent that's basically it now the properties of limits so we're going to suppose in each case that the limit of a subn and limit of B subn exist here's what we can do

with our limits first one if you have a limit with some constant times your sequence what you need to know is that anytime you have a constant next to your Your sequence or whatever your expression is in terms of one variable in this case it would be n you can pull that constant in front of your limit next one if you have a limit of two sequences added together we can separate limits by addition number three says likewise we can also separate limits by multiplication well if we can separate limits by Multiplication then we can

probably separate limits by division and we sure enough can and lastly number five number five is a really cool one I love this one it says that if you have a limit of some sequence to a power we'll call it P you can actually take that power outside of your limit as well so if you have a limit of a subn to the P power the whole thing's being raised to a Power this is the same thing as taking the limit of a subn and raising the entire thing to the P power so here's a

basic rundown on our properties for limits it says that we can pull constants out of our limits not a problem we can separate limits by addition we can separate limits by subtraction we can separate limits by multiplication we can separate limits by division we can do a lot of things of limits we can even if we have A sequence raised to a power we can take a limit of the sequence without the power and then take the the resultant after that after that limit to that power show F feel okay with the idea okay so

we're going to be using those uh I know that you've seen them before there is one qualification for this um a subn has to be positive and P has to be positive so we got to check for those as well now what I'd like to do is I'd like To go through some examples of how to tell when our sequences are going to be convergent or Divergent so here's what we're going to be going to this is just what we're going to be using some of the times to deal with these limits so the idea

is going back to this it says if you get a sequence let's find out what happens at the end of our sequence there is no end but towards Infinity what's going to happen with that you guys ready to do a couple Examples so we're going to try to show convergence or Divergence so number one let's say I give you a very simple sequence a very simple sequence like that one -1 to the N can you tell me what the terms of this sequence are going to be if I were to write out the terms like

we practiced last time what's the first term going to Be one or negative 1 we'd start with one right so 1 to the first power is 1 what's the next term going to be what how about after that and do you see it's going to alternate forever and ever and ever and ever and ever be negative 1 1A 1 1A 1 1 1 1 negative 1 now do that for eternity okay what's it going to be doing after eternity still wait is it still one it's going to one what's it what's it Doing it's going

1 1 one is it ever getting closer to one particular number no it's going 1 1 one and if you do this for eternity it's still going to be doing one negative 1 one negative 1 one is it ever going to one specific number no then this limit can't possibly exist this would be Divergent go one negative 1 one it's never going to a specific number does that make sense to you so this would say yeah no there's no way this is Divergent and that shows Divergence good enough with that one now for the other

ones this was very simple I just want to give you the idea that not everything is convergent and not everything when you take a limit of it this is one example of this not everything when you take a limit of it goes to infinity a number or negative Infinity that's not what happens this one doesn't go to a specific number this is one where the limit does not exist Does not go to Infinity does not go to negative Infinity it goes to one or negative one and over and over again it's not going to specific

number the limit just doesn't exist show F if you understand the concept there now for the rest of them well we have a much more concise way to do this without writing terms we don't have to write terms with these here's what it says if our sequence a subn is given by this 2 N / n +1 all we've got to do is take the limit Of that sequence as n approaches Infinity if the limit exists what do we know about that sequence did I not explain this let me explain if the limit exists we

know the sequence is okay so let's say we take the limit of this thing and it exists what do we know about the sequence good what if the limit does not exist or goes to infinity or negative Infinity that's the idea here so we don't have to list out all the terms that's not what this Says this says hey take your sequence here's your sequence find a limit as n approaches Infinity let's find a limit as n approaches infinity and if this is limit exists we know that we've just proved convergence I I need to

make sure that you guys are okay with do you guys see that this this right here is exactly what we're doing does take the limit of your sequence we're takeing the limit of our sequence this is our sequence right Here this would give us our terms right this would give us a first term second term third term all the way into Infinity we just need to do it once and for all we say okay well now let's let's find out what's happening to our terms remember that our this sequence is giving us our terms correct

let's find out what happens to our terms as we take our nend to Infinity that's the plan now how are we going to do something like that you actually could yeah you could Do L there's nothing that says you can't do lit talls with this uh first you got to verify something it's Infinity over it is infinity over infinity you could easily do a lowy talls with this not a problem you could also do some old school stuff could you divide everything by the largest power in the denominator you could do that so if you

divide everything by n if you divide everything by n we'd get 2 N / N we' Get N / n + 1/ n you can do It that way or you can do the L tall way since we've just done L this way it's probably going to be nice for us what's the derivative of 2 N what's the derivative of n + 1 what's that limit two notice that if you were to do this n / n is 1 n / n is 1 2 * 1 gives you two what's 1/ n as n appr

infinity so this gives you one uh sorry this gives you 2 * 1 this gives you one this whole thing is two as well remember this thing goes To zero either way you're going to get the two it doesn't really matter for that particular case so F be okay with this one now here's my question is this does this show convergence or does this show Divergence why perfect now does it say that this whole sequence equals to sequence is a list of terms you can't say it equals anything it equals what it equals it equals

all the terms up to the infinity Term okay now what I can tell you is that as we get close to Infinity those terms are all going to be really close to two that's what's going to happen so I don't know what it's going to what the actual terms are okay but I know they're going to be really close to two as we get towards Infinity does that make sense to you so what we all we're showing here we're not showing what this equals we're saying the limit exists therefore we have proven convergence for This

particular sequence and we just say that sequence is convergent okay let's do number three 2 +1 to the N Over N goodness gracious uh ladies and gentlemen how can we check for convergence or Divergence of a sequence what should we do take limit limit don't we love the limits no not really we can do them because we we've learned it from calculus one so let's go ahead let's take the limit of Our sequence what we know is that if we take the limit as n approaches Infinity of whatever our sequence is if this limit exists

we have proven convergence if the limit does not exist goes to infinity or goes to something like this we've proven that it's divergent or it have we've shown Divergence um ideas isn't anything with thetive 1 to the N automatically Divergent because no no it doesn't because What's Happening Here is because you're taking 2 + one or 2 minus one that's different okay so this particular thing this thing yeah it is doing this but when we combine it with everything else maybe it doesn't diverge maybe it is convergent we don't know we can't just automatically jump

to it so don't jump to it okay go through the process of actually finding out where the limit exists that would be a good good question but really bad idea to do that now is this Infinity over Infinity no no no it's not this is 2 plus or minus one over infinity okay that's that's that's way different so we can't use a Lyal rule all the time which is why I gave you this idea you're like why is he bringing that up well I'm bringing this idea up because sometimes we might have to actually do

that in our limits lals doesn't work all the time it's not our fail safe every single time you guys with me on that one so when you have something like man I have no idea This is something over infinity well uh to prove it if you're really kind of thinking about this you probably already have the answer you probably already see what's going to go on to really show it though you could divide everything by n the largest power in the denominator if you do that we get a limit as n approaches Infinity of 2/

n plus -1 to n/ n all over what's this going to be it's going to be one it's going to be n / N which is one now let's think about this how much is 2 over n as n approaches Infinity how much is that something thing zero now how about this one what's 1 to the N / n as n approaches Infinity this is also zero I don't care what that is whether it's one or NE 1 if I divide it by Infinity it's going to be zero does that make sense to you over

can you do Z over one no can you do one Over zero can you do Z over one this is 0 over one or zero that's what it's saying what it's saying is that as you're going to towards Infinity with your terms all of those terms are going to get really close to what number yeah I don't know what the terms are okay but I know that they're going to get really close to zero does the limit exist do we know the limit equals well yeah we do do we know the sequence equals no it's

it's a it's a sequence of terms okay we're Not adding them we're not subtracting them it's just it's a list of terms uh but we do know that that sequence is going to converge we have shown convergence show hands feel okay with that by the way I mentioned briefly that some of you if you're really thinking about it it could have shown that here that's a constant right that's a constant what's a constant over Bam could have probably seen it right there to show it mathema should be okay You can do this too but uh

you probably could have seen it have I explained these well enough for you one through three you want to try a couple more give you some different ideas here I kind of like these CU they go way quicker right now than some of the other stuff that we've been doing that take 35 steps to do it's kind of nice nice little break how about a sub n equal Ln of n n Oh goodness okay uh tell me the first thing that we should do in order to test whether this thing is going to be convergent

or Divergent what would we do take the limit very good so let's take the limit of our sequence where's our limit going zero I'm sorry where's our n going Infinity yeah we're trying to find out what happens as we get towards Infinity now we don't want to find out the first Finite number of terms because we could literally do that we could plug in the hundredth term and see what it is or the millionth term we could see what it is what we're looking for is what's happening as we get infinitely large we don't have

ever have to go to negative Infinity because n is defined as being positive right so positive integers don't worry about negative it's always positive that's nice for sequences we're always growing from one um tell me Something about the limit of lnn Over N almost sounds likeing lnnn n whatever lnn Over N uh what is it Infinity over infinity what we're going to use for Infinity over infinity in this case yeah use the L TS when you can I did I didn't do it here so I could bring back another idea but shoot I mean if

I was actually doing this problem for real I would definitely do ly TS absolutely anytime you have the infinity over infinity when you have those Polinomial and you can actually take a derivative or you have an Ln and you can take a derivative or some function you can take a derivative do it so we do a lital rule and we get the limit as n approaches Infinity of what's the derivative of how about thetive of how much is 1 over n as n approaches Infinity so this thing goes as 0 over one how much is

0 over one is this sequence convergent or Divergent convergent because because why Zer because the limit exist exists because it exists that's why so we just shown this sequence is [Applause] converged still okay so far let's do one more and then I'll talk about something else a couple of the ideas that we can use good ideas okay so we're trying to show whether these sequences are convergent or Divergent the first thing that we got To do is take a limit because we know if the limit exists it's convergent and if it doesn't exist it's Divergent

so number one thing we're going to try is let's do a limit as n approaches Infinity that's what we're trying to look at what's happening to our sequences we get way out there of this now we Pro I'm guessing that you probably haven't seen stuff like this in a while do you remember What to do when you have like a difference of radicals or a sum of radicals the conjugate that's a great word a conjugate would be an excent idea if we do the conjugate here um by the way we I don't want you to

just plug in infinity and go oh well it's Infinity minus infinity zero maybe it might be zero I don't know but you got to show it okay you can't take guess like that so anyway a conjugate is a really Good idea for this what it does is it makes it into a fraction for you and simplifies a lot of either the numerator or denominator depending on what you're conjugating do you guys remember the conjugate idea for Radicals now that we've seen it again yes we do the same exact radicals the sign is different that's the

that's the way that conjugates or if you want to be fancy conjugate uh that's the way just joking that's the way these conjugates Work that kills in my math C Class just so you know actually what is that really what it is just joking so when we distribute this thing uh notice that the reason why we use a conjugate is because we get this idea of a like a almost get different squares out of that uh the middle terms are gone so they simplify also a square root time a square root gives you the radican

so we get n +1 minus n over you just leave the entire denominator because what has happened Here is by multiplying by the conjugate we no longer have an infinity minus infinity we actually change that to an infinity plus an infinity and that we can actually do Infinity minus infinity I don't know infinity plus infinity that's definitely Infinity does that make sense to you also what happens with our numerators one we get one so we've get one we've got one over how much is this denominator for sure how much is one Over infinity Z actually

turned out that it was Zero but we had to show it um by the way if it equals z Z we got a lot of those do do the limits always equal zero obviously not but if they do equal a number what do we know about the sequence good since the limit exists we know convergence I want to Quick show hand you feel okay with with these ideas so use what you know about the limits to to find out whether the sequences are Convergent or Divergent all we got to do is take the sequence find

the limit of that sequence if it exists cool you know that sequence is convergent when we get toward towards Infinity those terms are all around one particular number if we do it and we the limit does not exist like that one that means as we go towards Infinity the terms are not all around one number they're either growing without bound decreasing without bound or they're alternating like like this One is they're not going to a specific number in that case we'd show Divergence of our sequence and that's basically the idea now we have a couple

other ways to do this if we can't for instance find the limit really easily we can use some techniques to not cheat but be really smart about showing a comparison of a couple limits comparison to a couple sequences maybe one way that we can do this is with what's called The Squeeze theorem You ever heard of The Squeeze theorem yeah you take this limit that's bigger you take this limit that's smaller you this limit that's right in the middle and you squeeze it and go haha sucker you're going to have to go to the if

both these limits go to One Thing the limit in the middle it's got to go to that one thing we can't have these limits go somewhere and this one go nuh okay it's got to go to where these go all right that's what the squeeze theem Does and I'm going to give you what the squeeze theorem says I'm going to show you how the squeeze theorem Works in just a bit so let's suppose that we've got three sequences we have a subn we got B subn and we got C subn do you notice that the

B subn is in between a subn and C subn for every single term this is what this means so this means that every for all these terms and all these terms sequentially This one's always going to be between them every single time so for the first term it's this scenario for the second term it's the scenario and every single time every single term until we get to Infinity it's always going to be a subn smaller than b subn b subn smaller than C subn does that make sense furthermore let's assume that the limit of a

subn equals the limit of c Subn and that those limits both exist that's why I'm putting l there that it's actual existence of those limits they exist does that make sense to you so far did I just lose you I don't L you here uh it says that hey suppose that this is the scenario we got the the sequence we're going to be trying to show right now is the B subn we're going to assume that we know what this limit is that it exists we're going to assume that we know what the limit of

this sequence is And that exists and that these two limits are the same so basically these two are going to the same number does that make sense they're both going to L well if that's the case if the limit of a subn goes to l and the limit of C subn goes to L and approaches Infinity tell me something about the limit of b subn as n approaches Infinity L it has to go to L it's it's squeezed between Them so again this says these two guys these are doing this a sub's going to one

number B sub's going to the same sorry C sub I misspoke a sub's going one number c subn is going to the same number B subn is somewhere between those guys right somewhere between there so if these are going like this to one number at Infinity this one's also got to go there does that make sense to you that's the way they squeeze there basically works So let me show you an example about how we can actually use it it's going to take us some time um do you have any questions on this so far

I'm going to leave this one to start over here questions on these ones are they straightforward to you mhm good okay I actually asked you for questions before I erased it this time learning you I'm going to call it uh yeah call as let's do that we got a subn equals n Over N the N I'm so excited factorials Super no it's not n n it's n factorial now here's the thing if we were to go through the same process that we just did the idea of finding out whether this sequence is convergent or Divergent

would be well uh you take a limit wouldn't you wouldn't you yes so we'd say hey what's the limit as n approaches Infinity of n factorial Over N the N and you think about it go okay uh as we code to Infinity we're going to have hey what's n factorial is n approach Infinity infity what's n to the N is n approaching Infinity infity no it's definitely INF Infinity to the infinity is going to be Infinity all right okay so we have an Infinity over infinity tricks me with this however when if we divide by

the largest power in the denominator we're dividing by n to the N right we'd get exactly the Same thing over again we get n plus n factorial over end of n over one that wouldn't help us if we try to do lal's rule this thing explodes this thing we don't know how to do derivatives of n factorial so maybe we have a better way if we have those factorials go oh my gosh I have no idea maybe try to do a squeeze THM on it try to find something that is smaller than this and something

that is bigger than this so here's what I Know I'm going to start off on one you know what I'm going to do I'm start off right here on one side I'm going to use the rest of the board to kind of work down to here the one thing I know is that this is always going to be positive would you agree with that always positive so I know right the bat that Z is less than or equal to my sequence that's what I know now the goal then is to find some other sequence that

goes right here that the Limit of it is what not one I'm trying to squeeze it right if I have if I'm going to squeeze something these two limits the limits of these sequences have to be the same so since I've bounded this below by zero oh zero it's got to be zero if the squeeze theem is going to work it would have to be do you understand the squeeze theem yes if I say uh hey tell you what um one sequence goes To zero and one goes to 100 what's this one go to I

don't know how supposed to know that I don't have no idea doesn't make any sense if I say this one goes as one and this one goes to one and this one goes to one that's how sque works so we're trying to go to the same number so if I say I know that this is positive therefore it's always greater than or equal to zero not a big deal I also need to find a sequence over here that the limit of That sequence goes to zero as well that's when the squeeze theem is going to

work for us you guys with me on this now in order to get started maybe we start by looking at the first few terms of what the sequence actually is doing we didn't have to do that with our previous examples but here it might be a good idea also I'll tell you this when I'm looking at terms of my sequence I'm not simplifying them I do not simp simplify Them and make it like one number over another number I leave them in their form that I'll show you right now I'll leave them in my form

that way I can see the pattern and maybe come up with some uh comparison so for instance a sub 1 gives me 1 over 1 to the 1 does that make sense yes I know that equals one but I'm going to leave it right here uh this a sub 2 gives me 2 * 1 over 2 2 * 2 2 the 2 power that's Right 2 the second power or 2 * 2 does that make sense to you a sub3 gives me 3 * 2 * 1 do you guys know what factorial means 3 *

2 * 1 over 3 3r right but that would be 3 * 3 * 3 does that make sense to you yes no yes no okay so here's here's the thought I'm gonna do something kind of tricky if I think about this I know that this one is always going to be less than Or equal to 2 * 2 over 2 * 2 isn't it and this one's always going to be less than or equal 2 3 * 3 * 3 over 3 * 3 * 3 isn't it you sure now one more thing this

is going to be kind of interesting here because the last number of a factorial is always one always do you understand that last number in the factorial is always one I can actually eliminate this last term and it's still right does that Make sense to you you can try it with a sub4 a sub4 would be uh let's see 4 * 3 * 2 * 1 over 4 * 4 * 4 * 4 I know that's less than or equal to 4 * 4 * 4 but look it the last number is one I don't

even need that over 4 * 4 * 4 * 4 here's what I'm trying to get you to see if I do a subn a subn would be n factorial Over N to the N well what I know is that this is equal to sorry less than or equal to the denominator I'm not Going to change but here's what I know about the numerator the numerator is always going to be less than n to to the not n to the n n the nus one I know that for sure um I get a different way

for you guys to see that too if you want if you want to look at at it differently so uh other if that just B did that blow your mind or were you able to follow that did you get that how many people got that feel okay about that did I blow anybody's mind like oh my gosh what in the world is happening okay um check it out let's think about what n factorial means let's think about what n the N means would you agree that this is always less than or equal to this one

always because this is n * n -1 * N - 2 time forever does that make sense but this one is n * n * n forever does that make sense to you are you sure now listen carefully if you Didn't get this the first time here's the whole idea okay here's the whole idea how many terms are here all the way to how many terms are going to be there Infinity not Infinity I'm going to go to n so if I give for instance if I gave you three how many terms are here three

if I give you four how many terms are here four four there's going to be n terms here there are they're n Terms n terms do you guys get me on that one are you sure how many n's are going to be over here same n there's going to be n terms now here's the part where we think a little bit do you agree that the last term over here is going to be one mhm okay so there's n terms but the last one is one one is going to be less than or equal to

any number in the world any number a whole number uh whole integer number positive in the world got qualify That a little bit uh basically it's going to be less than any one of these correct so because our last term over here is n and our last term over here is one I can take away this last N I can take away that last n and I'm still bigger than this because my last term here is one does that make sense to you so then I'm going to say tell you what this had n terms

but this would still be true if I had less than so here's n terms Here last term is one and then I did this where only have n minus one terms so I say let's compare these I know this is true I know this is always going to be greater than equal to this one obviously this is counting down this is 543 2 1 this is 55555 this one's always going to be bigger than that understood however I can be a little bit better I Can say well I know that if this is 5 4

3 2 1 that last one who cares I can do 5555 and it's always still bigger than that because the last term here is one does that make sense to you so I go all right well I think about this as n factorial but I think about this as not n the N how about n the N minus one so what this gives us is that n factorial is less than or equal to n to the please listen carefully so you understand where I'm getting this from This if this has n terms this is n

to the N does that make sense to you if this has nus one terms this is n to the nus one show P if feel okay with that is that a little bit did I explain a little bit better for you the second time around questions if you have them are you sure because I want you to get it before I go any further it's kind of an important idea I'm going to keep this on the board just work right down here all right well Then if this is true what we know about inequalities is

if I divide both sides by the same thing as long as it's not negative I still have a valid inequality since n the N is always positive I'm divided by a positive my inequality is still good we okay with this one now here's the cool part about this can you find the limit as n approaches Infinity of n nus1 n n yeah yeah we can do that actually um what we would do is we we would now divide by the largest power in the denominator if we divide by n to the n on both the

numerator and denominator we get a limit as n approaches Infinity of well this would be n n -1 Over N N All Over N the N Over N oh my gosh there's a lot of n's that was really stupid why did I do that that was really dumb I'm sorry I Was just going through my process that was that was ridiculous um we can we can do it a lot easier than that actually write that down edit uh what did I want to do oh yeah if you think about what this is think about what

n to the N is if I split off one n just one n then this is n nus1 * n do you buy into that one what do you do when you multiply common bases you add the exponents what's n -1 + 1 hey this is true then what's n nus1 over n nus one this is gone you get one how much is the limit of 1/ n as n approaches Infinity Z that's zero so let's put all of this stuff together what we know from here is that 0 was less than a to the

N uh a sub n we now know that a subn is less than well it was less than this this sequence n nus1 over n n because the limmit of this what's the limit of Zero zero what's the limit of this one because this goes to zero and because this goes to zero this says that the limit of a subn as n approaches Infinity goes to zero what's that tell you about our sequence is it convergent or is it Divergent it's convergent I can tell by some of the looks on your faces uh some of

them are like and some are like you just kind of gave up on the Problem I want to make sure that you get it all right uh I don't want to just leave you hanging on this stuff that's right leave you hanging on this de de stuff did you get the idea at all yeah what's whatsoever or are we just it's just processing stuff there a lot right there a lot of stuff that we just did we learned about the squeeze Thum squeeze Thum says uh these go here this goes here so we make it

up we go okay cool uh I know that zero I know that my sequence is going to be greater than Z I know that because I just looked at my terms look at this look at my terms this is n factorial right look at the terms please this is n to the N minus one I'm I'm stopping before the last one this is still true this is still less than this well that shows you that n factorial Over N to the N is less than n the N minus I'm stopping before the last term n

minus one hey hey look at that we got The same thing there uh n minus one I'm stopping before the last term because the last term here is one it's not make a difference what I have this is always it's still going to be a valid inequality the limit of this guy is zero you got it just showed it right there just showed it right here just split off the n n nus 1 * n this is still n the N these are gone we get one over n limit of that zero so I've shown

that Z goes to zero I've now shown that A subn is less than or equal to this a subn is less than or equal to this but this guy goes to zero therefore if 0 the limit of 0 is 0 the limit of this is zero therefore we now have a subn bounded between two sequences two different sequences but both of them go to the same thing zero goes to zero this limit or the limit of this sequence goes to zero therefore the limit of our sequence itself goes to zero and that proves that That

thing is convergent and that's the idea so fans feel at least a little bit okay about that one okay any questions all before I move on you see what I want to give you right now is two um two pieces of information that are very useful for you that we can use all the time from from here on out here they are any last last chance because it's all going bye-bye did you get this got that idea That's kind of cool right split that off okay so two notes with uh a couple examples very very

fast examples here this one is very cool we're actually going to use this later on as well in series uh but here's what you need to know if the limit of the absolute value of the sequence equals zero then the limit of the sequence itself also equals z you're like so what firstly why in the World what are you K are you kidding me here's what this is I I'll give you kind of a rundown okay look if you have a limit if you have a limit where you take the absolute value and equals z

if you have something that looks like this where it's going positive negative positive negative positive negative positive negative if you change all these two absolute value you get something that looks like This does that make sense down to zero so this says if the limit of the absolute value of sequence goes to zero then even though this sequence originally looked like this that's still going to zero that's what that says so the way we use it is with something like this one you okay cool let's talk about the limit as n approaches Infinity so suppose

we get this sequence -1 to the N / n suppose we're given that and we go Okay show convergence or Divergence you I don't even know because it's alternating right I I don't know well I mean you could just divide everything but or you could think about it this is alternating between one anda1 that's a constant number this is infinity it's going to be zero for sure but we can actually prove it with with this and this is very useful a lot of the times well just consider this one consider the limit as n approaches

Infinity of -1 to The N Over N in absolute value consider that one tell me what happens ladies and gentlemen with1 to the N if I put it inside absolute value what's it going to be it's not going be zero it's going to be one it says hey take positive one take the absolute value it's one take negative one take the absolute value it's one N I don't care what what we do with n because absolute value a positive number is always positive so it's really not for the n n is always positive does That

make sense to you always positive therefore this would be equivalent to the Limit as n approaches Infinity of 1/ n tell me what's the limit as n approaches Infinity of 1/ n absolutely and here's where our little theorem comes in it says check it out says take your sequence whatever you had take the absolute value of your sequence okay here's our sequence let's take the absolute value of our sequence if this Limit works out to be zero What's it tell you about this limit z z so the limit as n approaches infinity1 n / n

is also zero if the limit of the absolute value of our sequence equal zero AB sequence equals z the limit of the sequence itself also equals z show feel okay with that one that's real nice to get rid of thetive 1 to the N real fast you can do that a lot okay and we will be doing that a Lot okay last one we actually already used this very cool little theorem here it says if you have a continuous function and you have a limit of a sequence exists as n approaches Infinity then if you

have a composition so basically you have this okay you have a sequence look at look the bo here real quickly so I explain this you have a sequence whose limit exists inside of a Continuous function function we've actually used this before you will notice on the next example that we do we've seen this before we had e to the Ln we had we had sorry we had limit of e to some function we pulled the E outside the function remember we doing that I said oh you can do that because it's continuous I never really

proved it I never really showed it here's the reason why if you have a function that is continuous continuous and you have a Limit of a sequence that actually exists and you have a limit of well let's suppose that sequences inside a continuous function you can actually pull that function out do the limit first which also means that if this limit exists which is kind of cool all you have to do look at all you got to do is find the limit plug it into the function done let me give you a couple examples of

this this real fast cuz I do Not want to leave you hanging on this stuff and then uh we'll be done for today true or false the function e is a continuous function true or false true e is always continuous from negative Infinity to Infinity it just does it's always continuous because it it's continuous so e to X is continuous let's talk about the limit of s of 1/ n as n approaches Infinity please be Careful and don't rush off to something silly with S I know that s does this I know it does that

okay but look at what's going in what look at the argument of sign all right check this out as n approaches Infinity what's one/ n approach zero what's sign of zero do you did you guys get the idea here n is infinity therefore one over n as n approach Infinity is zero this is approaching zero what's sign as sign Approaches zero zero zero hey does the limit exist exists in fact it's convergent here's the cool part here's where the theorem comes in okay check it out it says uh the function is continuous and the limit

exists the function is continuous and the limit exists agreed that means that from here we could actually do e to the limit of sin 1 n as n approach to infinity or basically Think about this look at look at the board here real quick Z this limit is zero right therefore we would just have to plug this into the function e to the0 how much is e to the Z One isn't that kind of isn't that rad kind of cool okay last one and then I'll let you go promise oh you're right do it good

if you must go go uh I got to do this one because we're right here so if You got to go go you're not gonna offend me um but this is g to take us about maybe two minutes so first thing is tan inverse of X continuous yes no yes yes it is 10 inverse like this it is continuous can you find the limit of n^2 as n approaches Infinity okay this part is yes but what is it it's not a true question what is it inity Okay so it exists it's not convergent but it

does exist so this is continuous this exists here's what this says if I want to find the limit of tan inverse of n^2 as n approaches Infinity then all I got to do is find tan inverse of the limit and n approaches Infinity of n^2 now check it out okay watch careful this is the only only way I do okay tan inverse is continuous agreed that means that if I take a limit Of it I can pull that limit out in front sorry take pull that tangent inverse outside of my limit so it says this

is continuous this exists yes it's Infinity so let's pull that tan inverse up front what's the limit of n squ as n approaches Infinity again so what's tan inverse it's not a good math to write this but what's tan inverse of infinity what happens as we go towards Infinity what's what's this going to two that's right High over Two okay is that interesting or what wow did that make sense to you so P feel okay with what we talked about today good deal all right so the last little bit of our section here on sequences

uh the last little bit of our section on sequences talks about one specific type of sequence it's called a monotonic sequence and what monotonic means is a sequence and as far as a sequence goes is a sequence that is always increasing Or always decreasing so monotonic means you start with a one term of your sequence and it either goes higher and higher and higher and higher every single time or it goes lower and lower and lower and lower every single time either grows forever or declines forever does that make sense to you that's what monotonic

means the definition of it so a monotonic sequence is a sequence that is always increasing or decreasing a sequence that is always Either increasing or is always decreasing what would mean as far as uh always increasing or always decreasing in terms of our sequence it would mean this if if a sequence is increasing then if I have a sub one and I have a sub 2 and I have a sub3 and I have so on and so forth well if it's always increasing then a sub 1 has to be less than a sub 2 and

a sub2 has to be less than a sub3 and a sub3 has to be less than forever and Ever and ever a sub4 forever never never never that would be the idea of increasing it start off and then the the terms get bigger and bigger and bigger and bigger forever and ever and ever does that make sense to you okay decreasing of course would be the other way to say well you're starting off with some terms here but if you're decreasing then the first term would be bigger than the second term which would be bigger

than the third term bigger than the Fourth term forever and ever and that's a definition of the monotonic sequence it's a sequence that where the terms either grow or decrease increase or decrease forever and ever and ever should fans be okay with the idea of a monary sequence very good now what's going to be important for us is the act of showing whether a sequence is either always increasing or always decreasing we got to prove it because what we're going to be doing in Just a minute here's like the advanced organizer what we're doing we're

going to talk about monotonic sequences we're going to talk about whether or not they are bounded or uh sorry whether they're bounded or not um if we have a monotic sequence that is is bounded we can say certain things about that but in order to say that it's monotonic we've got to show that that sequence is always increasing or is always decreasing does that make sense we got to prove this or We got to prove this I'm going to give you a couple examples on how to do that then we'll talk about convergence as it

relates to monotonic sequences I just said a whole bunch of words did you were you following all that stuff so number one thing we got to do uh for monotonic sequences is show that they are increasing or decreasing for instance and I'm going to give you two methods on how to do this two pretty good methods so right now I want to Show that this sequence is increasing if you were to get this on your own and you well wait a minute I don't know whether n / n +1 is always increasing or always decreasing

you might have to plug in a couple terms and figure out what's going on so I'd plug in like I don't know one and I'd plug in two 3 four five and see what's going on here is this increasing is this decreasing turns out that these are these fractions are getting bigger and Bigger and bigger they are they're increasing here does that make sense to you okay so if we want to show this is always increasing there's basically two ways to do this one way to do this would be to consider start with the nth

term and if we want to show that this is increasing listen carefully to this because this can throw some people if they're you're really not with it right now okay if you're trying to show Increasing your first term or your nth term should be less than you could you could be less than or equal to uh less than or equal to your net next term now here's the question how would I get my next term n n plus one so for instance if this was my third term I'd have three and three correct that would

be less than my fourth term which would be four and four that would be 3 + 1 if I had my 11th term it would be 11 and 11 that would be less than my 12th term be 11 + 1 11 plus so basically anytime you're trying to do this compare it to your next term your next term is just well whatever you had because we're dealing with whole number integers add one that would be your next term over what would this be okay n + 1+ 1 I heard that n + 1 + 1

now of course we're going to make that a little bit simpler aren't we sure n plus two I show hands you feel okay with that as our next term so let's think about it Here's our nth term we got to be General right we can't plug in just one and plug in two because that would show that it's increasing only for one set of terms between a sub 1 and a sub 2 that doesn't make sense we got to show that this is increasing for all of our terms so for our nth term is it

still increasing from n to n + one so we say okay here's our nth term it's just our sequence the next term would be add one to the end well that's going to be the Next number past whatever you would say if you give me four I give you five you give me 11 I give you 12 I'm just adding one to whatever you're doing and that gives us our next term n okay to n+1 and + 1 2 n+ 1 + 1 or n plus 2 and if we're trying to show that it's increasing

this term should be bigger or smaller than this term bigger should be bigger than this term now if we simplify it just a little bit we've got This so basically we're trying to prove that this is true now here's some things that we can do with it if we're simply trying to prove that this is true we can manipulate these two sides to to make it mean something a little different or make it mean uh make it look a little bit nicer for us I should say something that's easy to identify what's happening because right

now like well I don't know is it obvious that this is bigger than this not really not super obvious Because I'm adding one to this but I'm adding one of that I I don't know so maybe we do things like um let's multiply both sides by n plus 2 let's multiply both sides by n plus one essentially we're going to take this and put it here and take this and put it there is that legal to do okay you have to be careful it's legal to do with inequalities here only because this is what type

of a number positive or Negative if you multiply both sides by a negative do you remember that you flip your equalities okay now think about what n is when we defined n n is positive correct n plus one would be positive n plus two would be positive so by multiplying by n plus2 we don't have to worry about inequality because we're multiplying by a positive number does that make sense to you multiplying by n plus one we don't have to worry about our inequality because we're Multiplying by a positive number so if I multiply both

sides by n + 2 and n + 1 respectively I get n * n + 2 is less than or equal to n + 1 * n + 1 if I multiplied by negatives I'd have to flip my equality because I'm multiplying by positives I don't have to do that show you Lo okay with this so far now this is this is a little bit better but let's go ahead and distribute and really see what's going on if we distribute we get n^2 + 2 N we get less than n^2 + 2 n + 1

it's bigger do you know that this is bigger than this side yes is it true for all n bigger than one yes so since since this this is this is kind of cool this is true for all n greater than or equal to one the reason why this comes about is because hey um where do our sequences typically start and they go forever right so we don't really care about zero we don't really care about negatives I don't care About that stuff I care about positive numbers starting at one because that's where our sequences start

so if I can show this for are you listening right now if I can show this for one and more then we're good to go make sense so if this is true well this came from this guy this came from this guy so because we manipulated this we didn't change around inequality we did all valid work with our inequalities here it says because this side is bigger than this Side it says this side is bigger than this side and that right there shows that we are increasing for every n that's greater than or equal to

one and that's all that counts because with our sequences we start with n greater than or equal to one and they're always positive show P feel okay with this so far so what we' just shown here we've shown that this is increasing this proves that this sequence is always increasing we were General we had NS we Didn't use specific numbers we're good to go you sure you're okay with that okay would you like a different way to do it yes so this true for all this that means that this is increasing sometimes not all the

time but sometimes we can use a very interesting idea to show increasing or decreasing here's another way to think about it so or I'll make a little caveat box four what if we had a different thought What if we said you know what I don't want to think of this like a sequence in fact if I just think of this not as a sequence but pretend that it is a function of X so basically instead of having n plus sorry n / n + one I think about a function of X XX + 1 where

I've got a function in terms of x got a function terms of X here's the interesting thought but firstly does this look something like that to you now you need to understand something this is Not this this is a function in fact it's a continuous function everywhere besides xal -1 understood it's continuous for one and more actually which is kind of cool uh this is not a continuous function this these are just fractions these this is a list of fractions that's all that this is so what we're doing here is we're we're kind of we're

kind of making this thing into a function to to find out whether it's in increasing or decreasing using this one idea back In your Calculus one class could you tell me when a function was increasing and when a function was decreasing and what did you use to do that derivatives and use specifically the the slope which is what what is the slope of a function in terms of calculus instantaneous rate change okay spec you're you're right it's inst instantaneous rate of change how do we find the slope or instantaneous rate of change it's the first

derivative in fact When you did uh it was one specific chapter for us it was chapter 3 it's called the first derivative test remember the first derivative test where you take the first derivative and you you and for us we would set it equal to zero and that would say where you're changing from increasing to decreasing and you pluged in two numbers it was positive if your first derivative was positive that meant your function was increasing do you remember doing that Your first Der was negative your function was decreasing for that specific interval and that

was nice we're going to use the same idea here if the first derivative gives us the rate of change if the first derivative gives us increasing or decreasing then we can use the first derivative here and here's how we do it if we did the first derivative now you got to know how to take the first derivative this is not a lall Thing okay this is a this is a derivative now we haven't done this in a long time how would you do the derivative of x x + low D High minus high D L

Square the bottom and away we are done I got mess that up not and away we go uh so low D High minus high D low Square the bottom away we go now of course I'm going to do this kind of quick right because we're supposed to be Really good at at derivatives at this point the first derivative therefore would be 1 /x + 1 2 so F you okay with with that one now you need to understand something about this a first derivative is a slope it is a rate of change is a formula

for the slope or the formula for the rate of change so basically if the first derivative is positive it means that our function is increasing if our first Derivative is negative it means our function is decreased are you all with me on this yeah here's my question look at this is this positive or negative or both both you plug in any number in the freaking world and tell me if it's positive or negative is it always positive forever for anything now we don't care about forever or anything be real careful here we care that this

is greater than zero or less than zero For X what X's do we care about do we care about negative X's right now no do we care about what is zero We Care X is greater than or equal to what to one now here's why this works for us just so you know we're we're taking this thing this sequence and calling it a function we're sfy the first derivative of the function if the first derivative of the function is positive it means the function is increasing if the first Derivative of the function is negative it

means the function is decreasing now we're saying xal 1 because really our sequence is going to start at one now we're going above and beyond here when you think about it this thing is not a sequence correct so I can plug in one and 1.1 and 1.3 and 1.5 and all these numbers in between 1 and two so we're basically going above and beyond what a sequence does let me make the point here for you If this works for all X's for X greater than equal to 1 it's certainly going to work for x =

1 and 2 and 3 and four and 5 and six and seven for when our n equals our X does that make sense or when our x equals our n for all the whole number integers show F if you see the the comparison here okay good so we're using the function to say hey if it works for everything it's certainly going to be increasing for all the positive integers and that's what Our sequence does as well so we use the function to kind of cheat at the sequence and say hey if that's increasing that's certainly

got to be you just can't call it an N because we're we're talking about a function here not a sequence so far so good okay so all last last time so for X greater than equal to one this is always true it's actually true anywhere but I don't have to have it anywhere I just have to have it for X greater than equal to one Because that's where our sequence starts so since this is greater than zero this shows increasing as well so there's really two ways to do it you can consider a term and

the next term and see what's going on you can consider a derivative of some function that's B based on your sequence and look at what's happening have I explained this well enough for you to understand it yes oh okay would you like to try another example yes Please so now on this example I asked you to show that that's increasing I kind of gave you a heads up and said uh tell you what this thing is going to be increasing you just need to show it well that that kind of gave us a clue as

to what to do here didn't it they said you certainly you're going to have determine the next term you have to have it anyway but we were looking for this we we manipulated it to see that that relationship uh this one well we could Have done that either way couldn't we if we just found the first derivative it would have shown up that that's positive would have shown up that hey if your first derivative is positive your function is increasing if your function is increasing the sequence upon which that function is based must also be

increasing for that interval well that means that if I give give you a function or a sequence I'm sorry if I give you a sequence and say just tell me whether This is increasing or not whether it's increasing or decreasing we might not want to start with this we might want to start with the derivative because that'll give us the answer typically every time now we can't always do a derivative I'll show you some examples where we can okay but if you can might be a nice way to start this so why don't you try

firstly to call this a function in terms of X again um we're using a function in terms of X And not in terms of n because n stands for our sequence it's the terms where n is based on one and two and three and four we're saying a function of X saying now this is going to be hopefully a continuous function for one and more we're looking for a continuous function here if that's the case if our function is continuous and our function uh is always increasing or decreasing then our sequence is so as well

but we can't call it in terms of n It's not the same thing so we are making a function in terms of X just basing it on the sequence what a function in terms of X look like can you tell me okay very good now what's the next idea what are you going to do say louder so I can hear you why are you going to take the derivative increasing or decreasing yeah a derivative gives you slope positive slope means increasing negative slope means decreasing go ahead and do the Derivative so quotient R my good

quotient room a while uh low D High minus high D low Square to the bottom away we go don't just take a derivative of the numerator and der denominator that's lall's Rule and that's not what we're doing here Mr Rob is that what you did low D High minus high D Square the bottom and away we're done way we go did you get that yes well that's really hard to see what's going On so maybe we start factoring and simplifying a little bit if we simplify this we can factor out the E to X can

you simplify the e to the X's this is gone the two is gone remember that uh e to x^2 is e to 2x that's where that's coming from so this is the same thing as 1 - x over e to X now we need to think about what is happening for the required domain of our function the domain of our function is X is What greater than or equal to one yes no yes greater than or equal to one why is it Greater why why do I have it defined as X is greater than or

equal to one are positive because our NS start where and they go higher 1 2 3 4 5 6 7 8 N9 that's why this is defined that way I'm trying to Define my my function as starting with my sequence because if this is increasing or decreasing for this entire domain the sequence is increasing or decreasing For every single n because all NS would fit somewhere in this domain make sense so let's think about it um what's e x is that positive negative or what if I now keep in mind if I'm plugging in one

2 3 4 5 6 7 is that positive or negative posi that's always positive so no matter what I plug in that's positive correct now think about this one 1 - x where X is 1 and more it's zero for the for one but then after that what is it then after that what is It after that what is it is it always negative so this is always less than or potentially equal to zero for X is greater than equal to one oh goodness that's so cool what's that tell you about this function is it

increasing or decreasing for all X greater than one yes yes makes sense hopefully it makes sense I'm teaching it shoot come on now so if this is always negative for X great equal to 1 then that function is Always decreasing for X than equal to one so that sequence is always decreasing for n is greater than equal to one you see the jump we just shown that function is decreasing all the time now here's the whole what we're trying to get to have I made this made sense to you any questions about this side here's

what we're trying to get to being monotonic is awesome I guess I'm always on the upshot so I'm monotonically Inc whatever Sucks to be a monic decreasing what a loser but anyway uh what we're trying to get is we're going to put two ideas together this idea of being monotonic and this next idea of being bounded a bounded sequence for some sequence if a subn is always less than some number if a subn is always greater than some number if a subn is always between Number two numbers what bounded means bounded means this idea that

the sequence is contained somewhere the sequence never goes above a certain number the sequence never goes below a certain number or the sequence is between two numbers forever does that make sense to you that's what bounded means so here's what we say um tell you what if every term of our sequence is less than seven or 10 or -4 or whatever it happens To be if every term in our sequence is less than a certain number one final fixed number we say that sequence is bounded above why because it's got ceiling you guys you guys

get get what I'm saying here it's got a ceiling there the ceiling is M this should be bounded above now let's say that our sequence I'm using uppercase M and lowercase M just so you know I I write in all caps so it's really hard for me but that's supposed to be lowercase M so If our every term of our sequence is always bigger than a certain number every term is always bigger than one or bigger than3 or bigger than zero maybe it's got a floor it never drops below a certain amount it's got a

floor it's bounded below does that make sense to you yes now what if both things happen What If look at this every term of our sequence is never smaller than this but never Bigger than this so basically it's bounded above it never raises above M it's bounded below never goes below Little M we just say it's bounded this would be a bounded sequence sandwiched yeah that's right it's got a ceiling and it's got a floor it's like us right now if I jump I'm going to hit the SC W my hands I can't jump that

high I I can jump but uh I'm not going to go below the floor right anyway Um I probably could get my head on that ceiling may run a little bit little Run start get a concussion that make all time uh views on all time views on YouTube that will get you views teacher gets concussion teaching calculus too all over Facebook goodness not going to do that so anyway we got this idea if our sequence is always less than certain Numbers bounded above if our sequence is always greater than a certain number it's bounded below

if our sequence is between two numbers so basically if it's bounded above and it's bound below it's bounded that's the idea feel okay with that one now here's the whole thing we're going to put these two ideas together let's suppose that we have a monotonic sequence this is going to be so logic I hope it's logical let's say we have a monotonic sequence and we Know it's bounded it's monotonic that means what that it's either increasing or decreas always going up or always going down and we know it's bounded always going up or always going

down and it's bounded tell me what's going to happen it's approaching a limit if it's approaching a li well we say the limit exists it's approaching the limit exists as we're approaching Infinity tell me Something about the sequence it's not mono decreasing or increasing yes it is it is it's just Bing it'll stop at some point no it won't stop what it's convergent I'll see you should have been louder I want you to consider what would happen okay look at look at this here's my bound above here's my bound below my upper bound my lower

bound are you guys with Me on this let's suppose that our sequence is bounded please please watch up here while I'm doing this okay suppose your sequence is bounded then your sequence is between here correct yeah this is not a monoton this is a piece of crap that's not nothing that's doing nothing for us but suppose that your sequence is monotonically increasing or decreasing I don't care what it is if it's increasing I can start whatever want as long as it's in here right it's going to be increasing can I do oh no no shoot

ah it's increasing but it's bounded but still increasing it's still bounded it's still it's still increasing is it going to a certain numberin what number it's going to it's going to the Limit as n approaches Infinity would be Big M does that make sense to You actually we don't even care about this one because it's monotonically increasing doesn't even matter that it has a lower bound here so every I I'll say a couple things every monotonically increasing sequence that's bounded above is going to converge every monotonically decreasing sequence that's bounded below is going to converge

does that make sense to you that's why this is here if it's bounded well if you're Monotonically listen if you're monotonically decreasing if you're monotonically decreasing you're going to be bound above at some point at some point because you you're decreasing all the time right it's never going to come back up so you can say it's fixed at that number not not an issue start with your first term that's your that's your bound above what we care about for decreasing is a lower bound don't we because we need we Need to have a floor we

need to have this the the planes falling right it's going to hit the ground eventually it's bounded below it can't keep going into the ground it's going to cause a crater a little bit that's horrible thought you evil people what are you doing putting thoughts in my mind but it's it's going to be bound below at some point if it's going to if it's going to land it's going to be bound below it's going to eventually get to one level that's what This is saying if we have a monotonically decreasing sequence and we don't have

this bound we can't say anything about it it's going to keep on going down we don't know but as soon as we say it's bounded below as soon we say that sequence is bounded then we know it's going to be convergent as soon as we say we have an increasing sequence and we say it's bounded above of course this is terms right as soon as you know it's Increasing if there was no bound we couldn't say anything about it but as soon you say hey it's monotonically increasing and there's an upper bound it has to

go to that number Big M show if that makes sense to you that's all I'm trying to say here so that's the idea behind putting this together let me give you just one example to illustrate this and uh talk about one more small idea and then we'll call it good for this section Show that that sequence is convergent oh my gosh how in the world going show that that that is convergent let's consider our sequence our sequence is 2 to the N given by the given by term by term 2 N Over N factorial so

far so good mhm the first thing you might want to consider is whether this thing is yeah whether it's increased do you know whether it's increasing or decreasing I don't know either well you Might want to plug in some numbers huh and find out whether it's increasing or decreasing um might not want to just start taking a limit arbitrarily so maybe find out whether it's it's going to fit this whether it's increasing or decreasing whether it's bounded above whether it's bound below let's deal with the increasing or decreasing first so let's talk about that uh

2 the N Over N Factor maybe we come up the first like five six terms and figure out what's Going on with this so we do first term would be 2 over 1 yes mhm next term would be 2 to the 2 over two okay well look at that is that increasing or decreasing same neither that's why for monotonically decreasing we had the idea of less than or equal to okay so it can be equal to it for a while but it's got to start going down eventually does that make sense okay let's do let's

do this is one This is two this was three what is it 8 over 6 oh something's happening this would be 16 [Music] over 24 24 yeah I fell asleep a little bit be 32 over 120 oh what's happening uh is this increasing or decreasing we're going to look to try to show that this is decreasing so if you don't know I don't know I don't know what that was you got to look at it I did because I did the problem before H uh but you'd Have to think about it what's what's happening here

this is staying the same and then it's getting smaller and smaller and smaller and way smaller and smaller and smaller and smaller so this thing right here says is this is decreasing however does this prove that it's decreasing forever we have not proved it we've done the first five terms so we have this idea that we think it's decreasing so it's Decreasing I don't know we're going to think about it we're going to see if it's decreasing now you got to show it so here's what we're trying to show I'm I'm going to change this

around just a little bit so remember what decreasing means decreasing means that a subn is greater than or equal to a subn + one that's what it means it means that as we go turn by term we're getting Smaller and smaller are you okay with a subn + 1 yes this means the next term so we have hey term number one this would be term two term number seven this be term eight term number 100 this would be 101 does that make sense to you the next term should be smaller now I'm going to flip

this around because I I want to show you a certain way so I'm going to flip this around another way you can think about it is that your next term should be less than or equal to your First term it's just a little backwards do these things say the same thing to you yes Okay the reason why I'm doing this is because you can show this one of two ways you can straight up just show it like we did one example or you can use a ratio approach watch this if I divide both sides by

a subn I would get a subn + 1 over a subn is less than or equal to if you can show that if you can show that you show monotonically Decreasing if you can show this you show monoton does that make sense to you monotonically decreasing means this that's what it means this is the same thing as this this is the same thing as this provided one thing happens tell me something about a subn for this to be true it can't be not I don't care about zero it can't be negative can't be negative a

subn would have to be positive otherwise your inequality would flip because you'd be dividing by Negative does that make sense to you so this will work if a and is positive if a subn is positive so shall we try I'm going to start from from right here so I'm going to do uh I'm going do this one so basically we're going to work a little bit little bit backwards we start with let's start with the nth term right now what we know is the nth term is this Guy it's 2 the N Over N factorial

you guys okay with that so far to the N or that's our nth term the next term someone on the right hand side you guys tell me what the next term would be how would you find the next term 2 okay hang on 2 n + 1 2 N okay two to two to the that bro two to the power perfect that's exactly right so look at if our if our first term is 2 the N our next term is 2 the n + 1 because we're dealing with positive integers here We're just adding one

to it does that make sense to you so if our nth term goes 2 the N our next term is starts out with 2 to the n + 1 so far so good what's the denominator going to Beal perfect D how we show that is with those parentheses and then factori quick show hands you feel okay with that so far so this is the a subn term this is the a subn + one term so far so good if we're going to try to show what we thought if we're Going to try to show decreasing

so the reason why we did the the first few terms we go well I don't know what this thing's doing let's find the first few terms the first few terms make it look like it's decreasing right that idea says okay if you're going to show decreasing this has to be true if you're going to show decreasing this has to be true if you're going to show decreasing this has to be true provided that's positive so if we're going to show Decreasing this has to be true let's make the jump stick with me if it's going

to be monotonically decreasing do you understand that this has to be true yes your second term must be less than your first term that's decreasing your second term must be less than or equal to your first term your next term must be less than or equal to your previous term that's all we're saying right here show F you okay with that so far now what's kind of nice about this is that We can use the division of fractions as a complex fraction to solve this thing pretty nicely pretty easily so check it out is this

always positive yes two then is positive yeah MH is this always positive yes for sure so let's use this idea let's say okay well now that we know that this is true if we divide both sides by our a subn we get 2 to the n + 1/ n + 1 factorial all over 2 the N of n factorial is less than or equal to 1 now That looks nasty but we're going to see a lot of this sort of stuff let's keep going with it first show hands you feel okay getting from actually getting

from here to Here show hands you feel okay with that that's good that's everybody that's fantastic now what do you do when you have a fraction divided by a fraction multip reci you reciprocate this and you multiply so this becomes n factorial over 2 to the n so Far so good yes no I'm going to kind of move a little bit quick on this stuff so stick with me here folks now think about what this means what this is I hope you guys are with if you get this you're going to get the whole thing

do you know that 2 the n + 1 is the same thing as 2 n * 2 because this would be to the first Power you add exponents when you multiply common bases This is n factorial no this is 2 to the N no big deal with there here's less than or equal to 1 now here's the cool part I want you to think about n + 1 factorial n + 1 factorial would look like this please watch carefully it'd be n + 1 time n time n -1 * N - 2 * forever until

you get to one does that make sense to you this is n +1 factorial that's what that is is that clear for you you start with 11 then you do 10 976 here's the cool thing about this if this is n n +1 factorial what's this n factorial it's n factorial therefore here's a little side note for you keep this in the back of your mind all the time n +1 factorial is the same thing as n +1 times n factorial does that not make sense to you it does hopefully it makes sense to you

n factorial * n +1 is n +1 factorial this says hey you taking n +1 * 1 less Than that n times all the stuff before n that's n +1 * n factorial you guys okay with that one well here's what's nice about that then this thing is n + 1 * n factorial do you think see anything that's going to simplify for us do you see why we do you see why we did that what what simplifies factorial bam gone by the way do you do you understand why we didn't just do a derivative

here do You know how to do derivative n factorial no okay how about this one two what do we get 2 question you need to now say whether this is true or F false for n greater than equal to 1 is this true yes for n greater than or equal to 1 yeah plug in one what do you get you get one that's true statement plug in two three and four and five and Anything bigger than one is this always going to be less than one yes then this is true for n bigger than or

equal to 1 therefore we've shown that this is always decreasing we start we started with decreasing didn't we look at look at the board please we started with decreasing right hello right yes start with decreasing we showed that our work is all valid this is always true for n bigger than equal to one so therefore This sequence is always decreasing for n greater than equal to one if it's always decreasing that's monotonically decreasing you guys okay with that so far now let's consider what's going on here so 2 n/ N factorial is number one monotonically

Decreasing number two I want you to think of 2 the n/ n + 1 factori sorry Over N factorial tell me something about 2 N Over N factorial tell me something think about what its smallest value could possibly be could it be negative no okay so right there you got it if it can't be negative what's the smallest value you could possibly attain or get close to zero that means it's bounded below by zero does that make sense to you it's got to Be you answer the question yourself if it can't be negative then this

must be bounded below by zero this is because a subn is always positive now I'm going to make a little caveat here this is bounded above by something okay it's probably bounded above by where we start so are you with me if you plug in one what get when we plugged in one uh we got it two or something we got two so The upper bound is two now do I care about the upper bound right now this function is always decreasing sorry the sequence is always decreasing always decreasing I care that it's decreasing do

I care about the ceiling no do I care about the floor yeah so I care that it's decreasing and bounding below I would care that it would be increasing and bound it above it's not increasing though it's decreasing is it decreasing and does it have a lower bound then what Do you know about the sequence every bounded monotonic sequence is is this bounded yes is it monotonic yes don't care whether it's monotonic increasing or decreasing if it's bounded appropriately this is decreasing bound below therefore this is convergent bam now you know if that made sense

to you you sure yes do you get the idea what increasing is what monotonically Decreasing is do you understand that if it's increasing and bound it above it's going to converge don't pack up yet I have one one more minute to uh to do stuff I'm going to do it too it's going to take one minute so do you understand that it's bounded above and it's increasing it's going to converge if it's bound below it's decreasing it's going to converge so basically and basically if it's if it's monotonically increasing there will be a lower bound

As well that's why they use that word bounded so it's bounded increasing no problem bounded decreasing no problem that's the case we have here the last little bit is just a little statement um it's a very kind of obvious statement I want to give it to you before we end because it'll help you on some of this some of this stuff little note I want you to think of a number to a Power of n so like you give me a number give me a give me a number one what say what one one okay

one or eight or 1/3 and I say okay now take that number and take it to the first Power then the second power then the third power then the fourth power then the fifth power and six power and seventh power and every positive integer forever okay I want you to look at what those numbers are doing when are the numbers going to converge When are the numbers going to diverge you said one if you give me the number one it's going to be one forever isn't it that would converge if you give me the number

negative 1 it would be 1 negative 1 1 negative 1 one or actually netive 1 onega 1 1ga 1 onega 1 forever would that converge or diverge if you give me eight give me eight 8 to the 1st 8 to the second 8 to the 3D a to the 4th what's happening monotonically increasing Without bound well that's the whole thing right it's got to be bounded that would diverge how about one half one2 would be decreasing it would also be positive that would be bounded below mon the decreasing and bounded below by zero that would

converge you guys with me about negative one2 no no negative2 okay this is going to be weird But2 to the N same thing as1 to the n time 12 to the N yes no remember that whole idea that if the absolute value of the sequence converges sequence converges do you remember the whole idea about if the absolute value of the sequence converges then the sequence converges boah converges okay so here's the point there's two numbers that are kind of our guidelines one and 1 anything between oh do you understand That anything between there is going

to get smaller fractions take fraction to a power what happens I don't care that it's negative it's going to go positive negative positive it's going remember that whole thing it's going to do this remember that yeah it's going to do that if it's negative if it's positive it just does that if it's a fraction that's what that says the reason why the negative 1 is not included is because the negative 1 we already did negative 1 To the N goes negative 1 1ga 1 1 1 one it never gets any closer one's there because it's

always going to be one does that make sense to you okay uh so also last thing you can already tell what this is if the limits going to exist what's it going to be zero it's either going to be zero or one yeah if it's monotonically decreasing or it's the absolute value is is going to tend Towards one number it's going to be zero for R is not equal to one you know I'll say it this way for R between those two numbers it would equal one if R equals 1 that's the only two options

you got this is either going to go to zero or it's going to go to one if this is one naturally it's going to be one to the whatever power is going to be one if it's not one it's either going to diverge if it's outside this range or Going to converge if it's inside that range have I made that make sense for you yeah okay question okay