Calculus 2 Lecture 9.2: Series, Geometric Series, Harmonic Series, and Divergence Test

So now that we've just finished talking about sequences we're going to progress one little step and talk about series you see with sequences we just stff like this we said suppose we have a sequence of terms and it goes for a while and we looked basically at whether this thing with this sequence would converge or whether the sequence would diverge and we I I showed you in last Section basically how to do that remember we taking a limit of it working with a limit maybe using the squeeze theorem sometimes doing the comparison tests with that

well now we're progressing a little bit further if we have a sequence and we say well now I want to do something more than just look at the terms as being a list what if I wanted to actually add them together as soon as we take that sequence we go no no no I don't want to Just list them out I want to add them what we get is called a series now there's a couple different ways that we're going to look at series the first first notation that I'm going to introduce to you and

you've seen this before it's probably been a really long time since you have seen it Sigma yeah it's with that Sigma notation so when we use this Capital Greek letter Sigma uh Sigma s stands for a sum so whenever we see that we have a sum so we say here's Okay suppose that we have remember that notation what would that stand for that would stand for if I did this would be a sequence no now it says Okay cool so I want you to take these terms of your sequence and I want you to add

them together from where I tell you to start to where I tell you to end that right there would be a series are you guys okay with the notation of a series quicke not if you are okay with that so Here's what this would do this would say okay here's terms of your sequence right boom boom boom boom boom so boom boom boom boom this says you add them together instead of just list them out it tells you where to start and it tells you where to stop where do we start one where do we

stop we do not stop we keep on adding them and adding them and adding them and adding them now there is another way that we're going to consider This we can look often times as we're where we're going to work with a lot today for the next probably 30 minutes or so we can work with what's called a partial sum can look at what's called a partial sum I want you to consider the comparison of these two things firstly would this still be a series yes where would it start yeah would start at the first

term when I plug in one for K notice how I'm using a different uh Different index here I'm not using n I'm using k because where does this series stop that's right so here's what this would do if I say no I don't want to start at one and go to Infinity I want to start at one and go to 100 or a million or something finite well this would add up all those terms now here's the here's the point I'm trying to make here do you understand that n is going to be your stopping

point so because n is finite Here then what we're going to get is something that that for sure exists because this is a finite number of terms because we start at one we're going to add these things up we're going go hey what's the first term cool what's the second term cool what's the third term cool what's the nth term cool add them all up is that for sure going to be there yeah if I say hey um here's some series and add up the first 10 terms you can do it Can't you first 100

terms you can do it can't you the first n terms you can do it can't you the first Infinity terms I don't know I don't know what that might be because that that goes forever right but as soon as I say you know what n's a finite n's a finite number here so because n is finite n is finite that's a finite number of terms this has to be some sum so let me go over that here real quick Because I'm going to I'm going to use this information just a bit so we have this

thing called a series what a series did does it says hey you know what it's no longer just a sequence it's no longer just a list of terms now we're going to add those terms together that's what this stands for generally speaking this is what we're trying to do here folks we're trying to start at our first term and add up all the terms for Infinity this is what we're trying to do now Because I can't just add things forever cuz I'm not going to live forever uh probably you never know uh but because I'm

not going to live forever I can't add them forever so what I'm going to do is I'm going to try to work around this idea I'm going say you know what wait a minute let's not go forever let's go to a finite number now because I'm starting at the number one for my terms here so the first term and because I'm stopping after some finite amount I'm going to Able to add those terms together does that make sense to you this sum has to exist because n finite subn your sum your partial sum this is

called a partial sum here your partial sum must exist show fanty if you understand that concept okay now here's the important part about this you understand this finite right this can be a number like 10 or 100 or 200 do you understand that for every additional n so if I go from 1 to 10 That's going to be a number right if I go from 1 to 11 that's going to be a different number if I go from 1 to 12 that's going to be a different number 1 to 100 that's going be a different

number do you guys get the idea well then what's going to happen little subnote here for I'll call it the next n for the next n so the one right after this one Remember these are U by the way just so you don't lose track of what we're doing here that n that still has to be positive it still starts at one and and it's still just your inte it can start at zero sometimes but it's just your integers all right so it's not like uh any real number they are positive integers here you with

me just like on sequences we're basically just taking a sequence and adding up the terms okie dokie so for the next end that means hey Stop at 10 now I want you stop at 11 now I want you stop at 12 so for the next n we're going to get a new sub subn we're going to get a new sum here's the interesting thing I'll try to put it all together for you all right put it all together sequence if I don't add them series if I do add them this is the way we write

series what we're trying to do is figure out this sum the sum of this series now the way we're going to do it We're going to look at a partial sum we're going to say cool cool let's go from one to a finite number because we're going from one to a finite number this sum has to exist now for every next n that I get I'm going to get a different partial sum does that make sense if I get a different partial sum then check this out this is pretty cool if I just go from

one to one I'm going to have partial sum one to two partial sum 1 to three partial sum 1 to four Partial sum one to five partial sum so for every additional N I get a new SMN this creates a sequence of partial sums do you get that idea that's kind of cool right that's weird that's why we cover sequences this creates a sequence of partial sums well let's think about the sequence of partial sums this is so nice the notation works for us if I go from one to if I stop at one look

at that n would Be one n would be one I would get s 1 if I stop at two I would get S Sub 2 if I stop at three I would get a sum of the first three terms if I stop at four I would get a sum of the first four terms and so on and so on until I get to wherever you tell me to stop I will get that many different terms does that make sense to you this is our sequence of partial partial sums now here's the main notes for us

we'll get into some examples in just a Minute uh this is all like the the theory behind it our our base okay this is our foundation so here's some notes for us why we did sequences is for the next couple statements that we're going to make you see because I'm going to a finite n yeah that's got to exist but because every n would give me a different sum that creates a sequence for me the first sum the second sum the third sum the fourth sum the fifth sum The nth sum well because that's a

sequence here's a couple things we know what we know is that if the sequence of s subn converges if we know the sequence of sub subn converges then the limit of sub subn will have to exist so this by definition our limit of s subn would equal let's call it s as n goes to Infinity well then this is what this is going to show for us I'm going to I'm going to show you the the notation here In just a minute but if your sequence converges check this out this is pretty cool if your

sequence converges that means that this sequence is going to one specific number correct that's what conversion means in fact that's what this means right it says hey look at that your sequence as n goes to Infinity is giving you a a specific number does that make sense to you your sequence is going into one number that's what what was our Definition of convergence here Well what are the s's what is s subn it's a sum it's actually a sum of whatever this thing is right now now stick with me here this is where we make

the the jump the like calculus jump because this uses limits Okay so do you guys understand that our s subn or S Sub one that's a sum s sub2 is a sum and sub3 is a sum and S subn is a sum now check this out if your sequence converges the limit of your sequence as N approaches Infinity would exist that means that your s subn term as n approaches Infinity is going to be there that means your sum as n approaches Infinity will exist if your sequence of partial terms is convergent if that converges

what that means is is that if this happens okay one more time s subn is a sum right if this happens you know where the sum is that's the idea does that make sense to you because this says hey you know what uh Stop at 10 stop at 12 you're going to get sums aren't you now don't stop well you got to look at your sequence if your sequence converges you go okay cool go to Infinity your sequence is there that sequence converges that sum exists that's the sum of your series that's the idea you

just keep on extending it one term at a time one term at a time until you get to infinity and then if your sequence converges that means that that the uh the limit of your sequence gives You that one number that one number is your Su let me write some of this notation out we'll go over one more time I'm still getting some glossy looks from you kind of glossed you over there let me show you some notation I'm guessing it's going to make some more sense okay so I'm going to throw a limit up

on the board here real quick uh show you how this relates back to your limit of of your partial sums and then we'll Hopefully get to it you guys ready let's start with this so we have a sum or a series of a subn from n 1 to Infinity here's the idea the the idea is dang I don't know what that's going to be so here's what we're going to do we're going to make this equivalent to a partial sum but the only way we can do that is do this this is the important part

okay check it out so here's the idea you go okay uh let's consider this instead of going from 1 to Infinity let's change our index here let's go from K = 1 to n of a sub K here's my question to you let's see if you're having calcul if you have a calculus brain today all right here's your here's my question are these equal right now no why not you need to take so would be a finite number wouldn't it are do they start the same place yeah and these terms would be the same this

would be based on K this would be based on N but no big deal you get the same numbers Out of it all right however this one don't stop this one does stop doesn't it so how can I make this expression equal this expression how can I make this equal liit where would n goity perfect calculus brains you got calculus brains today that's good now these look the same to you yes this says hey you're going from one to n but now take n to Infinity that creates this for us well here's the cool thing

about this we already have a definition For what this thing is this right here is your partial sum do you see it okay so stick with me here folks stick with me here's the jump here's another jump okay so because this a partial sum because we're going going to from one to a finite number yeah we're going to take the N to Infinity later okay this is what we're doing later because we're going from one to a finite number you said this thing has to exist remember saying that over here you said This thing has

to exist all the stuff says that well then this equals s subn you guys with me on that one this part is s subn does that make sense now check this out we've already talked about this this is a sequence this is a sequence S Sub 1 s sub2 s sub3 all the way to s subn in fact we've talked about this if this sequence converges if this sequence converges this is going to go to a finite number S if that sequence doesn't converge then it doesn't go to a finite number s doesn't go to

a specific number it goes to infinity or negative infinity or something alternating like we Hada 1 one something that where the limit does not exist however if this does exist that means that well let's let's read it back if the limit of our sequence s subn exists that means that the limit of our partial sums exists the limit of a partial sums is our series and that's What we we just made it down to so if feel okay with with this statement so basically it says all right uh got a series Infinity call it a

partial sum that partial sum I don't care what it is this partial sum right here has to exist bam we already said if I take a limit this may or may not exist I don't know yet I don't know okay but if it does exist if it does it says here's what's going on you have a sequence of terms Yes this is a sequence of terms those terms stand for partial sums they stand for the sum if I take the sum at Infinity that's what this stands for this is the sum at Infinity well if

that converges that means it goes to a specific number that would be our sum of our series you guys with me I hope I explain it well enough for you to understand did I do that for you okay some people get all like oh my gosh how does this how is this a sequence why is It relating well it's related because you have a sequence of terms which are your partial sums because we can go to Infinity with a sequence we can go to Infinity with our partial sums we can go Infinity now with our

series would you like to see an example example yes a simple example whatever tongue twister it was a tongue twister last thing just so we know this maybe it's an obvious statement to You if the sequence F subn diverges what's going to happen to our series typically what's going to happen if this diverges this goes to infinity or negative Infinity you add up to something that goes to infinity or negative infity sometimes you'll get the just diverges the limit does not exist therefore you can't do it example time do you have any questions on this

whatsoever because I'm going to erase it we're not going to talk about anymore This is all the theory behind it we're just going to do examples now so do you get the idea of a series that terms being added together yes no yes do I get that we can find a a partial sum no matter what because when I take a limit of a partial sum it goes to my series therefore if my sequence of partial sums is convergent my series is convergent if the sequence of partial sums is Divergent then my series is Divergent

that's basically it What's also very cool is that when I do my limit I actually get the sum which means I'm add I I've added up that series because that sum is going to Infinity hey sorry because the sequence of partial sums is going to Infinity when I get there that is my sum of my series cool it is cool I think it's cool it's very interesting to me time too I just wanted to make sure you see the interplay between the sequences and The series because some people are like well we did sequences now

series are completely different well we're using sequences in our series here so example n goes from one to Infinity of n something very simple here's the idea of what you're supposed to do the first thing that we're supposed to do I wish I could have left this on the on the board but this is good enough The first thing we're supposed to do is we need to find a partial sum because ultimately here's what we're doing we're going to try to do this whole thing right typically what we'll show is we'll show this to a

partial sum and if that partial the limit of the partial sum exists we're good to go so we got to find a partial sum now that is not as easy as I just made it seem uh sometimes they are very difficult to find in this case this One's going to be fairly straightforward but this is not always the case so s subn that's that's the def that's our our notation for a partial sum can you tell me what the partial sum would look like 2 okay so the partial sum says take this and then start

adding up the the terms in there the terms adding up the terms of a sequence so the terms of our sequence would be one yeah and then two and then three and where would I stop very good I'd stop at n should P feel Okay with that so far do you understand that I'm stopping because now it's a partial sum I'm going from one to N whever I stopped this is our partial sum here so S Sub n's yeah okay start at one go to n we've just done that now step number two this is

where it gets difficult this is typically pretty easy you just plug in some numbers you go and then you ultimately plug in N no problem step number two is you got to Find a formula for this that's the hard part oh sorry uh yeah find a formula for for for estim find a formula fortim so basically you need to do this what that equals what the formula for the partial sum is not just the terms and the sequence okay look at look look this is the formula for the terms in the sequence it's done okay

it says that hey you want your 10th term in the Sequence it's 10 hello you should plug in 10 that's easy what I want now is a formula for the partial sum a formula for this crap well does anybody know what the formula for that is n n close no we're Downs sliding oh you're closer with the first one uh you actually use these in your ran sums if you remember those I think it's it's story that gaus uh found this out when he was like in elementary school or something pretty Cool was that genius

uh he he's super smart was he's really now he's dumb right CU he's dead he can't do no more math maybe I don't know maybe wherever he's at wherever he's at so this is our our our formula for the partial sum s subn so what we know is that s subn is equal to n n + 1 / 2 you guys okay with that one so far Okay so what we've done is all right here's our here's our series no problem it says series adding it up says start with a start with a list of

your your terms of your Series so we're we're doing okay S Sub thenal sum would go from one plug in one two plug in two plug in three all the way till you plug in and you actually have to have that nth term in there do not just leave this hanging with your dot dot dot okay end it with your nth term do you got me yeah Now it says okay cool find a formula this is the hard part so finding this formula is the hard part the rest of it's very easy so once we

find a formula awesome here's what we've done we've just done this part we we have that now now it says I want you to determine what's going to happen as your end goes to Infinity so basically if the partial sum for every term that you want is given by this formula check it out if I said hey find the partial sum uh for the Six for the first six terms you should have to plug in six it'd be 6 * 7 over 2 done you guys with me on that one well that that's awesome now

it says find out what happens as n goes to Infinity so our step number three is really pretty straightforward step number three is just take a limit as n approaches Infinity what this does is this makes our partial sum equal to our series that's what that does check this is why I asked you this very first question I said all right now check it out are these two things equal are these two things equal right now with me covering this up no no they're not so this would not this would equ this but this would

not equal that correct cuz these two things are the same so if these two things aren't equal the way I make them equal is I let this go to infinity and that's where our limit is so as soon as we find our F Subn just take your limit as n approaches infinity and then if it converges awesome we found our sum if it of our series if it doesn't converge well then we can't find the sum of our series it's impossible to do that quick head now if you're okay with that fun right so step

number number three says no problem just take a limit as n approaches Infinity of n n + 1/ 2 how much do that get to equal don't Even worry about this about doing a lot of math on this how much is that going to equal that's going to be Infinity that's infinity plus one right that's Infinity what's infinity time Infinity definitely Infinity over two cares Infinity what's that tell you does the does the sequence here here's perfect terminology I hope that you're getting this is the sequence of partial sums convergent is the sequence remember these

are partial sums right is this is A sequence of partial sums is a sequence of partial sums convergent no it's Divergent what that means that when you get to the infinite term the sum of that n equals infinity you're you're going to get positive Infinity here well then what's your sum it's Divergent we don't know because if our sequence of partial sums is Divergent that means our series is also Divergent so we'd say series all that work for Nothing series is Divergent I know we got very little payoff with that one but I wanted to

show you that not all series are Divergent but this is the way you go about it so ADV feel okay with with that one good good all right can we move on to another example do you have any questions at all on this one before we get going yes am I explain this well enough for it to make perfect sense to you of course first time for everything right All right so uh let's do more I'm going to try to give you a whole bunch of different types of examples here because a lot of these

things are done different oop sorry I'm so used to X's um what should it be yeah for some reason I really like these type of series some people don't so much I do so what I'm going to try to do is I'm Going to try to see whether this series is convergent and if it is I'm going to try to find the sum if it's not then we're good to go we're done so I'm going to try to find the sum of this series what's the first thing that I might want to do combine FR

find the partial sum find the partial sum so I'm going to try to find the partial sum so the partial sum says okay well the first thing that you better do I'm I'm going to write a little bit smaller Than I normally do because I'm going to need some space here first thing that we do is we well we list out some terms here so we we better list out some terms how do you find terms in a series what would you do yeah because what a series is is just a sequence added together so

let's plug in one if I plug in one I'm going to get 1 see see where the one comes fromus one perfect this would be n = 1 that's our Nal 1 term you okay with that One what's the next thing I put nope before that oh plus a plus sign because it's a series okay so let's keep on going what's going to be our next two fractions can you tell me that 17 - 15 very good this would be n = two what's the next thing I do plus very good okay can you come

with two more fractions here for me n - 17 1 nth - 1 7 very Good I love patterns this is our third term show P you feel okay getting those those fractions so hey you know what you plug in one you got it you plug in two you got it you plug in three you got it now you need to do this until the nth term so basically yes you're looking for a pattern okay you're looking for for something you can make a formula out of we're going to skip a whole bunch of them

and we're going to go to oh what do I want to do yeah I'll do That for you I'm going to go to ah shoot are you guys going to get super confused if I go down here and do dot dot dot and then plus down here no Okay so line just use your imagination I want the Nal nus 2 terms I want the N = nus1 terms and I want the n = n terms actually you know what by WR uh because it's a partial sum we could use We could use this we could

use K really that's what it should be uh because our partial sums are K to n so if you really want to be mathematically prudent here you would do K and then we do oh okay cuz Nal nus 2 that doesn't really make a lot of sense we do K and we do K and we do k um are you okay with that one reasoning behind it is because s subn the the series from uh k equal 1 to n that was our S Sub n okay that gave us An actual sum here so so

really when you're doing the partial sums you should have your letters in terms of K Are you seriously though no pun in 10 are you okay with that okay a okay a okay now the last one's I'm going to work backwards here the last one's really easy if I let these go to Just n I'm going to have 1/ 2 n + 3 - 1 2 N plus one do you see where that's that one's coming from mhm now if I have to find these guys this would be 1 over it Wouldn't be 2 N

- two bless you yes nus 1 + 3 very good - 1 over 2 N -1 + 1 notation getting sloppy there we go show fans if you understand that concept you feel okay with that one guys in the middle yes no are you sure so this says hey you're plugging in two plug in two plug in three plug in three plug in N minus one you're plugging in N minus one Here it is nus one and N minus one that's all you're doing make sense now if I want n minus 2 it would be

what one over what 2 - 2 - 1/ 2 N - 2 + 1 ladies and gentlemen are we all okay with that so far if you didn't simplify your fractions now's where you'd want to do it okay if you didn't simplify these things now is where you'd want to do it So I'm going to write these in purple uh this is going to be just my my my fractions that are simplified so we [Music] have one over 2 N that look at that if you if you distribute that and you combine some like terms

how much are you going to get 2 n - one 2 nus one you with me are you with me yes this would be 1 Over 2N do you see the minus three that one come on folks help me out here one distribute combin like terms one over 2 and what is itus one over please don't you give me 2 N on this minus one do you see where Theus one comes from you can't just subtract one add one you have to you com my terms now this one this is good to Go we fine

okay quick show hands you feel okay with this so far would you be able to do this on your own of course you would it's pluging in numbers this says plug in one plug in two plug in three you got the fractions this says plug in I go backwards on these ones all the time I plug in N N's easy man exactly what you got plug in N plug in N minus one plug in nus 2 and simp are there any questions on how we got those this is Important that you understand how we got

those do you understand how we got those are you sure yes okay so I'm not going to look here anymore I don't want to look there I'm going to look at these fractions I'm going to find out what in the world is happening here Charles you okay um the second one should that be on the bottom one n or 2 nus one for this an this yeah so -2 - 2 + 2 I think we're right I think we're Good sorry no problem you okay with this now obviously we don't have every single term listed

because we don't know what n is ultimately n is going to go to Infinity that's why we had to include the N you can't just say let's go to seven okay well no cuz the limit would't work there right well what's awesome about these types of telescoping series you should be writing that series is that a lot of times most of your terms they're gone check it out now when You're Crossing these out you need to follow the pattern please listen carefully what I'm saying follow the pattern it's crucial do you see anywhere else where

you have a positive 1/5 and a minus 1/5 look at the relationship amongst the terms here's how I do it first fraction crosses with the last fraction and the next term so first fraction last fraction next term did you get that first fraction last fraction next term did you got that first Fraction it would cross out with the last fraction in the next term gone well we've done the we've done the first one let's work backwards now okay let's go backwards here let's let's look at this um last term our last fraction crosses out with

the first fraction in the previous term do you catch that one yes no okay well last fraction crosses out with the first Fraction in the previous term that was the same pattern only in reverse first fraction cross out with last fraction next term first fraction crossed out with last fraction next term now here's the issue first fraction would cross out with the is this here no so this stays there's nothing there we stopped at N I don't need the N plus1 term we stopped right there does that make sense to you mhm Now look at

this one is this one going to cross out MH our last fractions remember this relationship last fractions crossed out with the first fraction term previous last fraction would have crossed out with the first fraction the term previous it would be gone here's the last one how about the negative 1/3 last terms cross out with the first fraction in the term previous is is there another fraction before this so then basically this is so Awesome what are the only things that are going to be left in our partial sum 13 oh no it's not 1/3 13

and 1 2 3 do you understand that every other fraction in this problem is gone they all gone they gone they're all gone everything else gone well hey look at that right now we have a formula for our partial sums this was step number Two we created a formula for our partial sums isn't that kind of nice if I say I want the partial sum after the 10th term here's what you do You' plug in 1/3 you plug in 10 done done that's it now the question becomes does this converge or does that diverge what

do you have to show for that take the limit just take the limit this is now your formula so in Step number three we go all right well let's take a limit of our partial sums as n Approaches Infinity that means that we'll take a limit of our partial sums as n approaches Infinity can you tell me ladies and gentlemen what's going to happen to my limit of this partial sum as n approaches Infinity no the whole thing zero what happens to this that's zero what happens to this here's the awesome part about this this

goes back to the theory that we talked about the very beginning not The theory uh the work that we did basically your partial sums are going to eventually end at- 1/3 does that make sense and they don't really end but uh they converge to 1/3 well because this is equal to our Series right now we know the series is equal to 13 that is awesome let's it's very kind of interesting that when we add up all these terms forever we'd still get- 1/3 Jessica you had a question yeah could you explain how the 1 over

2 nus 3 got crossed out again 2 n- 3 yeah this one okay so did you notice how the 17th that's a last fraction right yeah it's Crossing out with a previous fraction like a previous term the 1/5 is crossing out with the previous term the this one would be Crossing out with a previous term just like this one did just like this one did okay now we don't go any FR than this that's our nth term so this one doesn't get to get crossed out this one doesn't Because there's no previous term okay that's

the idea we just pretend like we have a ter dot dot dot this right here is called a telescoping series telescoping series happens like this very frequently which is very nice for us because we can find a formula so this is a telescoping series tell you what let's do one more just so you get a really good idea about what I'm going to be asking you for here some of you might not like this very Much but uh telescoping series really only work if you have a sum or difference of two fractions typically a difference

between two fractions do I have two fractions here no no no no I don't uh now if you think of wait a minute why don't we just take a limit of that thing remember that this right here is a sequence it's just a sequence and then when I add them the terms together I get a series so I can't just take a limit of This here's the limit would tell me limit would say uh the sequence of these terms exists because it goes to zero the sum is not zero the sequence converges therefore the series

is probably going to converge here but we got to show what the sum is to find out what the sum is we find the sequence of sums and we take a limit so the the work is not over just by taking a limit all that did for us was say whether the sequence is going to converge I know the Sequence of these terms would converge the sequence of the partial sums I don't even know because I don't even know what this the formula is for the sequence of partial terms now because we know that telescoping

series usually works with two fractions uh I might be able to try to write this out and find a pattern to it for this this the um for the series but often times it's easier to try to create something like that a telescope in series so what we're going to do We're going to take this 4 over 4 n^ 2us 1 I want try to factor that thing this would be 2 n + 1 2 N minus one you know what I think it does work out better if I do this easier to see let's

do minus one plus one to minus1 2 N + one does that look familiar to you difference squares it is difference squares so we should know how to absolutely factor that guys with me on That one does this look familiar T turn it into two fractions how do I turn into two fractions equals two two okay sure two that seems right to me hey we are geniuses does that work no the common [Laughter] denom do you know what those numbers are I know you can do it but I don't know What those numbers are oh

crap what's that it never leaves you you're never free you only think you're out of the Matrix too bad we're not in linear algebra you really would be in matrices that would be awesome you never have the Matrix right okay so anyway I'm going to go through this quickly you should know how to do partial fractions but understand that what a partial fraction will do for you here is decompose this thing into a couple fractions maybe Those things are telescoping hopefully for us they will be so we would have 4 = a * 2 n

+ 1 + B * 2 nus 1 he I show hands feel okay with our partial fraction here remember that the cover up method you go okay A 2 n + 1 b 2 nus1 and then we start plugging some numbers in here's how I would do it I'd say let N = -2 and let N equal pos2 the reason why I'm doing that remember we plugging in those those uh Numbers with which make one factor zero that's what we want to do so if I have - one2 here this whole thing goes to 0

do you see it I would get 4 equals if I have -2 here I get -2b n = -2 I plug in 12 for N2 in 2 * 1/2 is 1 1 - 1 is zero this whole thing's gone 1/2 here 2 * 1/2 is 1 1 + 1 is 2 4 = 2 A or a = 2 what that means for us is that now instead of thinking of that series I just decomposed my nasty Fraction there into something that is hopefully telescope so series of 4 over 4 and n^ 2 - 1 is

equivalent to saying the series N = 1 to Infinity can't change that uh what would our a become what is that now two two two over what plus oh minus why minus over does that look a little bit better Like what we've been dealing with so far what's the next thing you're going to do find the partial sum partial sum very good so step number one is partial sum how do you find the partial sums what do you do plug in some numbers let's start where do we start one if I plug in one I'm

going to get oh good I'll leave two over one minus what okay let's plug in two two over what Minus am I going too fast with my adding here 2/3 okay on to the next one let's sometimes two good okay let's do next one two k remember this would be k k = 3 so 2 fths plus- 2 hopefully 7th do you guys see where those are coming from they're odd numbers uh this is a formula for odd numbers so was that one so you just plug in up hey look at that uh 1 3

35 57 blah blah blah blah blah blah blah blah blah do you guys get this Yes yes no yes now I'm going to jump down here again I want the K = nus one terms I'm not going to do the nus 2 this time I'm going to do the k equals n terms because this is a very straightforward very straightforward series when you're doing this yeah start off with one start off with plugging in two start off with plugging in three I always end with plugging in N first because it's easy to do if I

plug in N I Get 2 over 2 N -1 - 2 over 2N + 1 you guys okay with plugging in N it's just the ending terms of whatever this series says so F be okay with with that one now let's plug in the N minus one I plug in the N minus one we got two over 2 * n -1 - 1 - 2 / 2 N -1 + one that's 2 over 2 nus 3 Perfect 2 over 2 okay listen I know I did the fractions very quickly um what I'm asking is do

You understand this idea of plugging numbers in getting it say specifically these two terms so dep hands if you do understand that okay now this one's this right I'm not going to worry about this anymore I'm worried about this look for the pattern what happens please this is the important part you guys can do this right you can do you can do partial fractions you can do that you can do this it's plug in numbers in plug in one plug in two plug in three plug in N Minus one plug in N stop there now

look for the pattern in your own words what's the pattern last first cool that works would this one cross out no there's no last would these cross out yes these ones this one okay how about uh let's see where am I at this one yes this one yes yes no yes n is the one that doesn't n oh you didn't put that bro that so inside I I called them insides I Don't know why insides insides insides insides insides last one nope there's no next one so there's no first one there's no so yeah outsides

whatever so we'd have what's left I've got a two I've got a - 2/ 2 n + 1 what's the last step folks and what's the limit of this Z two two minus zero sure but uh don't forget that you have a two right there show hands if that made sense to you guys middle people are you guys okay With this are there any questions on this right now so let's continue our talk about these these series now the next thing that we talk about in our series is specific forms of series uh so far

we've talked about sequences how to tell when they're convergent or Divergent then we kind of use that idea of sequences to tell whe When series are convergent or Divergent and find the sum by doing those partial sums remember taking a Limit of a partial sum gives us whether that series converges or diverges do you guys remember what we did last time hopefully well one specific series that we're going to get a lot is this thing called a geometric series what a geometric series is is this series of a very specific form like that where you

have a some constant times R some base raised to an N raised to expression in terms of n where n is the uh the index that's going 1 2 3 4 5 6 7 to Infinity so this isn't verify that's a series firstly look at the terms of our series the terms of our series would start with with what one well it wouldn't be one it start with a if I plugged in one I'd get R to the Z which would be one but then a * 1 would give us a and then would add

to it what would add to it what would the next term be a a r very good what would the next term Be sure and then the next term would be ar to the 3 and it would never ever end because this thing's going to Infinity should be okay with with that one very good okay now there is one other small change that we can do with this you could have it this way or you could have and you can verify this is the same thing or you could have a series from Nal 0 to

Infinity of a r to the N can you verify That those two things are the same series for me please yeah if I plug in one hey I start with r the 0 if I plug in Z I start with r the 0 it's the same exact thing so this will give us exactly the same thing what I need you to know is that both of these Expressions stand for geometric Series so far so good now I'm going to prove something in just a bit but I want you to know a couple things before I

prove it first thing geometric series Converge when you probably you know what you could probably think critically about this and you could you could probably tell me geometric series converge when a is basically meaningless as far as convergent or Divergent it's a constant it will not change anything about this you can actually pull the a all the way out front and be fine with it and multiply later okay we're going to find out in our our properties at the very end of the section that we we could Actually pull a out and be fine it's

a constant tell me something about R these series are going to converge when R is what less than one a kind of a big number for us as far as our series go imagine if R is one if R is one then I'm going to go a plus a plus a plus a plus a plus a for Infinity if I do that forever I'm going to end up with infinity it's basically infinity times a Constant does that make sense to you if R is bigger than one R is bigger than one I'm going to go

a then something's going to be bigger than that and something's going to be way big then way bigger way it's going to go to Infinity if it's positive or maybe negative Infinity I don't know but it's definitely going to diverge because my numbers going to go bigger or way smaller and bigger than way smaller if it's alternating or something like that So here's a deal this is going to converge when R is less than one in terms of absolute value what that means for us we can think about it another way RS got to be

between two numbers for it to converge what two numbers do have to be between yeah if you didn't think it's not just simply less than one it's actually between negative 1 and one because if you had an R being Like -4 do you see that if I had R4 when I Square it's going to be huge or what not Square huge but when I Square it's going to get really big when I Cube it it'll get really big negatively and then when I take the fourth power be become positive again does that make sense to

you so the whole thing about geometric series is well if you're going to add up these terms the only way that they're going to add up the only way that you're actually be able to find a sum is if These terms keep getting smaller and smaller and smaller basically it's got to be decreasing does that make sense to you in absolute in terms of absolute value it must be decreasing well the only numbers that do that the only numbers that that decrease when I take them to larger and larger powers are fractions that's the only

ones that do that and so what this says is that our geometric Series yeah they can converge but they can only converge if the value Of R is less than one not equal to one we already talked about equal to one right if they're equal to one you're going to keep on adding a forever it's going to grow without bound absolute value less than one is the same thing as RS between 1 and one also what's very cool about this oh you know what I can say this little correl here it will diverge if R

is greater than or equal to one in Terms of absolute value what's also really nice is that we have a built-in formula for the geometric series I'm going to prove this to you just a bit but I'm going to give it to you now here's our formula if you know that you fit the form of a geometric Series so basically if you know you have a constant time some number raised to the nus one where your R in absolute value is less than one you can find the sum of It and the sum's really really

nice uh so here we go the sum is a over 1us R that's it you can always find the sum to these so converges when absolute value of R is less than one Divergence absolute value of R is greater than one would you like to see the proof for this why this actually happens okay let's do that so here's what I'm going to do I'm going to start with just my general geometric series Here and what we learned last time was that if I want to find out what the sum is or if I want

to find out whether the series converges I'm probably going to be looking for a partial sum first remember talking about the partial sums last time look for a partial sum first so I'm going to look at SMN my partial sum well what my partial sum is going to be is this we already actually have a partial sum up here already we got if I start with um help me out again with This if I start with one what's my what am I going to get and then I if I go with the next term I'm

going to get and then I'm going to do a r s and then I'm going to do a r cub and I'm going to do a r now here's the deal a partial sum ends a partial sum ends at n does that make sense to you ends at n n so here's the deal listen carefully if I plug in N will my last term be a * R to the N will that be my last term no because if I plug in n i hey you know what I plugged In one I get zero I

plugged in two I got one I plugged in three I got two I plugged in N I get n minus one naturally hopefully you see that this has to be my last term it's got to be a * R N1 show hands feel okay with that so far awesome now I'm going to do some kind of Fancy Pants math here what we're going to do is we're going to create another expression and then subtract these two things you're going to be a little bit fancy about it so what I'm going to do I'm going to

multiply because you do believe this right this is a partial sum well it's an equation to which means that I can do whatever I want as long as I do it to both sides I'm going to multiply both sides by R so I'm going to have R * s subn equals If I multiply both sides by R understand I would have to distribute to every single one of these do you guys see that I distribute to every single one that means I'm going to have a r plus a r 2 + a r CU plus

the Very last term what would the very last term become when I multiply it by R very good I'd have one additional R notice if I multiply this by R it be a r n -1 * R to the first you add exponents when you multiply common bases that would be a r to the N power quick head now if you're okay with with that one now here's what we're going to do now this is kind of cool you believe this expression right just a partial sum no problem we Actually had some of that on

the board we're just stopping because you say partial sum goes up to a certain value of n that would be n we get our our ending term here this is a partial sum multiply both sides by R we get this one and now we're going to subtract sub ract these guys if I subtract them what we end up getting is s subn - R * s subn equals so I'm subtracting this minus this I'm going to erace these RS here basically We're just multiply by R that means you got to multiply this by R okie

dokie with that one now we're subtracting s subn minus r sub r s subn got it that means we're going to have to take this stuff and subtract all of this stuff what's going to happen with most of these terms yeah in fact what's the only couple terms that don't cancel a a doesn't because there's no a down here so a the r okay squared cool boom boom a Doesn't and what other one doesn't what about that one is that one going to go away yeah that one's going to be here somewhere does that make

sense that's going to be here somewhere because what would happen is I'd have the a sub R sorry a * R the N minus 2 and I multiply by n that become n minus one and that would go away with this one so this is gone I'd have this one now would be a negative oh yes I'm sorry uh I wrote Plus yeah we are subtracting good call good catch I would have caught it about 5 seconds but I saw the saw that I don't know why so yeah we'd have this minus this okay well

that this a stays minus 0- 0us 0 minus 0 but then subtract that last one Patrick's right we get minus a * r n show P feel okay with that so far now here's what we're going to do what we're going to do right now is we're going to solve this for sub subn what that means is we're going to factor if I Factor out my S subn I get 1 - R if I factor out my a I get 1 - R to the N still okay with that one last step let's just divide

by 1 - R are you so far so good on this now here's the interesting thing we started with a partial sub of our geometric series I've worked it out mathematically to say this is still a formula now for the partial sum of our series basically all we've done we've Made a formula out of this does that make sense to you made a formula well here's what we learned last time last time we learned that if we want to see whether this series converges what we would do is we take a partial sum and we

do what with it limit limit as approaches what okay if you're if you're confused back the video up or or go and watch this video again um here's what we learned last time we learned That if you want to find out whether a series converges find a partial sum a partial sum ends at n correct but then what you do is say hey now let the end go to infinity and then your partial sum translates to your series if you let this the partial the limit of a partial sum is n approach to Infinity is

your Series so this is what we had I'll write everything else for you this is equal to the limit of your partial sum that's this Was last time okay this is not new stuff this is true this says if you find a partial sum for this guy and you let that n Go to Infinity that's true however we already know an expression for this I don't want this one that's nasty right I can't do a limit of that that's that sucks what I can do is a limit of this thing though so now that we've

worked down to hey your partial sum is this let's just take a limit as n approaches Infinity of a * 1 - r n over 1 - R now I want to think really carefully about this what is changing and what's going to what's going to be a constant in this particular case firstly tell me what a is Conant a is a constant R the number R will be fixed depending on what your series is so it would be something like 1/2 something like 4/3 something like seven whatever negative 3 3/4 whatever that is your

R will be fixed you understand that the only thing That's moving here is your N N's going towards Infinity now here's the whole point about why this is the way it is I want to give this to you first that way you understand here what's going on okay let me let me full attention ention here if R is one if R is one this would be 0 over zero that would be a big problem for for us we would have a limit that does not exist therefore we would get diverges when R equals one does

that make sense to you it's clear that it Will because if you plug in one for all these things you're going to get a * Infinity there you go now let's say R is more than one if R is more than one so like a number like two what's two to the infinity it's Infinity that's where this comes from if you're are an absolute value is more than one you're going to get something that is infinity here when's the only time in the world you can take a number and take it to the Infinity and

it's going to give you zero less what's less one when it's a fraction that's what this is coming from okay it says hey you know what if you take a fraction less than one so this equals z if absolute value of R is less than one that's where this is coming from I'm going to try to go through it one more time for you so you really do get what's going on do you get what's going on already a little bit what's going on is That this is our partial sum no matter what we have

for R does that make sense to you I've proven it right here this is our partial sum for our series no matter what we have for R however if we have R for certain values we're going to get two different outcomes if R is one or more we get Divergent because this thing is going to Infinity are you with me we get infinity over constant number because R is a constant Infinity over constant is infinity it's Divergent if we have R less than one though in absolute value so between 1 and one between them well

then that number raised to the infinity well that's going to go to zero 1 - 0 is a * 1 is R is a constant here we'd have a over 1us R and that is proving that proving two things here it's proving two things three things really it's proving that this series diverges when R is more than or equal to one in terms of absolute Value it's also proving that the series converges when R is between 1 and 1 so absolute value of R is less than one and it's also proving that the sum is

equal to this because when you think back to the last last thing we did in this class last time it said that when you find out the limit of your partial sum your your partial sums are a sequence and the ending sequence would be when ends Infinity the ending term of your sequence is still a sum it's the Sum at Infinity so we we said this if your limit exists your series converges more than that because you're finding the limit of a partial sum if your limit exists it also is the sum of your series

it's also what this thing equals and we just prove that show hands you feel okay with that one now do you want to see some examples yeah you're going to like these you really are promise [Applause] Now in order to use this stuff you first have to determine whether this thing is actually a geometric series or not so we're going to look at it is it of the form of a geometric series yes if it is and here's how you check here show you check your first exponent should be zero it should match up with

one and N minus one or zero with n does that make sense those are the two options that you have in this class so 1 and N minus one 0 and n It should have a constant if any it should have some number being raised to that power so in our case we should have an A you should be able to tell me what the a is and what the r is so tell me what's the a please three three very good does the a have anything to do with convergence or Divergence no tell me

what the r is does the r have anything to do with convergence or Divergence oh heck yeah that's why we did this whole thing so You tell me is this geometric Series going to converge or is it going to diverge how do you know it's going to converge that's right the r is between Nega 1 and 1 or the absolute value of R is less than one not a problem so because we have just proven this what we know is that this series converges did we have to do any work with that nice H series

converges and you should be able to tell Me the sum the sum is this is the sum right here all we got to do is plug the numbers in guys it's that simple what's our a again over it should be one minus r be careful on your signs don't screw this part up 1 - what perfect 3 over 1 --2 she be okay with where those numbers coming from now can you do 3 over 1 -2 yes sure that's how Much three over three Hales [Music] or sure 3 over one reciprocate multiply we get two

so we know a couple things it's kind of interesting right we know a couple things firstly I know because this series is geometric and we already prove it already proven it because our R is less than in absolute terms of absolute value is less than one I know it's going To converge more than that I can plug my numbers in uh make sure you don't use positive 1/2 here you will get a different sum okay you do have to plug in what the actual R is plug that in here's our a here's 1- R figured

out than that we know our sum is two that's really all we got to do with geometric series what if this - one2 had been a four3 would that be a convergent Geometric series or a Divergent geometric series why what if it was negative 4/3 would it still be Divergent yes absolutely the absolute value of 4/3 is still 1 and A3 it's still more than one so in those two cases we get something that would be Divergent we wouldn't have to now listen here's the whole thing okay here's the whole thing if you find that

your geometric series is Divergent should you find the sum no heck no it's Divergent notice though that the formula will let you find the sum won't it you can still plug in three and you can still plug in -4 third and it will still give you an answer is the answer going to be right no so don't do that okay you first show convergent then you find the sum you do not do it the other way around you don't find the sum and say hey the sum is this but the series diverges that does not

make sense at all Okie doie so show convergence then find the sum if it's appropriate does that make sense to you so this one this right here this would be diverent therefore this would make absolutely no sense to do let's move on sometimes we got to do a little bit more work with these guys I'm going to show you one of those let's say we don't have an explicit A subn but it's not given to us all we know is that this thing Is 5/3 plus I'm sorry minus 5 9ths plus 527 - 5 81

plus so on so forth so it's alternating alternating signs we know five is always on the numerator and we're going 39 2781 I'm going to show you a couple ways that we want to work with this number one thing notice where our series is Starting where's our series starting at one or at zero starting one what that means for you is that my exponent has to be n minus one does that make sense to you you got to match it up it's got to be zero with n or it's got to be one with n

minus one now if you you just heard me understand that so I'm going be going to find an expon of nus1 for my R now how we do that well try to factor and simplify as much as you can to find that pattern so for instance I would Factor out the five I would also factor out the 1/3 from this and here's the reason why what I want is for my first term to be one Okay the reason is because I'm going to have I'm take something to the zero power does that make sense so

if I leave it as one if I leave it as a five and then a 1/3 does something to the Zer power give me 1/3 no but if I factor Out the three as well does something to the zero power give me one then yes that's what I want does that make sense to you I'll say that one more time when you're finding your series if you start at one you're going to want to have an exponent of n minus one that's one thing we need to know second thing because I'm going to have an

exponent of n minus one my very first term inside my parenthesis should be one because when I do 1 minus one it's going to give me zero for my power anything to the zero power is one that's what I need have happened now I'm going to be a little bit more pattern oriented for my next part of this instead of like something like - 1 27 I'll think about this as 3 3r instead of 1 over 9 I'll do 3 the 2 instead of 1 over 3 I'll do 3 to the 1 instead of 1

over one I'll do 3 to the AHA do you see what I'm talking about about the first Power has got to be zero This is what I needed if I would have left this as 3 to the 1st I would have had a little bit of a problem here cuz there'd be no way to for me not a real good way for me to reconcile I start at one but I'm getting a zero does that make sense if I start at one with a one well I'd have I'd be off by a negative if I

did start at zero with a sorry one with an N I'd be off by my sign so we do got to factor out that 5/3 not just the five well here's what's going on then What's going on is I have 5/3 and I have look at this this is to the 0 power to the 1 power to the second power to the 3 power this is 1/3 to now we're very close let's see if it works firstly okay let's see if it works I start at one right so I need nus one power it's got

to match up if I do that if I start at one my first term would be at0 13 zero then I get 1 over 3 to the 1st then second then third are you guys Okay with this so far now here's the problem I'm alternating how do I fix this a negative here if I put a negative right there well shoot let's make sure that this works okay 5/3 would be there no matter what correct then my series would go through if I plugged in one I would get 1 - 1 is zero I would

get one out of that that's here if I plugged in two I'd get 2 - 1 is one I'd get my negative back I'd get my one back I would get my three to the first back I Plugged in three three I would get 2 - 1 is sorry 3 - 1 is 2 I would get a positive I'd get a one and I get 3 squar you see see that this is our pattern this is our our series now well that's kind of cool because now I know that this thing equals this thing 5/3

13 to the nus1 can you tell me now what type of a Series that is go ahead and you discover if it's convergent if it is convergent find the sum if it's not then say it's diver stuff right there go for it I so if this thing is a geometric series we should probably be able to figure out whether it's going to be convergent or not just by looking at it is this going to be convergent yes or no yeah it's Geometric it's a geometric series with a what's our a what's our R gooda 13

is that betwe negative 1 and one then this thing is going to be converted that's the first thing you say okay this converges then we find out the sum the sum of this series is how would you find out the sum sure a over 1us R did you guys do that as well 5/3 over 1 - 1 thir how much is that all said 53 54 so our sum is 1 and 1/4 how many people found it are you guys okay with this so far easy medium hard pretty easy as long as you can get

down to here right once you get to here I mean this is a piece of cake this is awesome just don't do silly things don't find out that and then find things like sums that Would be silly find out whether it's convergent or Divergent first and then if it converges find the sum let's do one more you see sometimes these things aren't that obvious that they are geometric series like that one does that look like a geometric Series right now no heck no but sometimes you can manipulate it well enough now here's the here's the

thought process you need to focus on this one because this might leaving the dust if you're not okay Geometric series match up Nal 1 to a power of n minus one you get that or be Nal Z to a power of n now if we know that then a lot of times we can manipulate our series to see if it's going to be geometric for instance here's what I would do instead of e to the N I'd try to force this thing to have an exponent of n minus one can you verify for me that

e to the N is e * e Nus1 okay now this is 3 n + 1 1 what I know is that 3 n + 1 is 3^ 2 * 3 n -1 can you verify that for me yes no yes okay because when you multiply common bases you add those Powers right so if I had 3 2 * 3 N - 1 it' be 2 + n -1 as my exponent that would be 2 + that would it would be n + 1 that would be my power there so F feel okay with that

one guys over here are we okay with that you sure okay man hey um quick question can I Separate fractions by multiplication yes multiplication is this a constant that's e over 9 this is e over 9 times e n-1 over 3 nus1 tell me something about e nus1 over 3 to the N minus one what could I do with that maybe I could make one fraction to a power couldn't I because they both my numerator denominator have the same power then this thing is the same thing as a series N = 1 to Infinity of

okay e Over 9 that's fine it's ugly but it's fine and then I'd have e over 3 to the n-1 now we are practically done with this problem but I want to walk through it one more time to make sure that you guys get what we're doing here first thing we're doing we're checking to see if it's geometric we're just we're looking at saying hey can I make it into something that's geometric if I can I need to look With where my n starts if my n starts at one I need to match this up

to a power of n minus one did you guys get the first step yes no then try start doing that say well wait a minute then e to the N no no no I don't want to N I want 2 the nus one let's split off an e so if I go all right well e to the N I don't like that I want to think about this as e to it's close but it's not as e to the N minus one * e to the first because if I Add those together I still get

e then now of course I have written backwards but this is the idea try to find out the appropriate power from our Expressions here e * e nus1 no problem same thing with 3 n + 1 1 well 3 n + 1 is 3 * 3 * e * 3 N - 1 that's 3^ 2 this is a constant pull that off this I can rewrite that because they have the same power as e over3 nus1 and all of a sudden we have this thing that is a geometric series with an a of what's our

A e over9 it's weird but it's e over9 what is our R now you tell me is that going to be convergent is that going to be Divergent you better know how much e is before you make that statement off it's about 2.71 or so so is this going to be convergent or Divergent yeah hey look at this this is less than one not a problem so this is going to be convergent if we want to find our sum Our sum would be a over 1 - R well let's do a little bit of fraction

work we'd have e over 9 over if you want a common denominator that's 3 over3 - e over3 that's going to be e over 9 3 - e over 3 reciprocate and multiply we get e over 9 * 3 over 3 - e if we Simplify we get e over 3 * 3 - e it's just some fraction work but that's how we do it that would be our sum right there it's a weird sum but what I'm asking is is that a constant is that a number yes sure could you find out an approximation

on your calcul L if you really wanted to you could do that uh ladies and gentlemen how many of you feel okay on getting down to this Step At least how about determining whether this is a convergent or Divergent Geometric series can you do that do you see where all this junk came from then you then you know all you need to know okay the next thing rob you have a question the next thing we're going to talk about as a different type of series we're going to talk about the harmonic series most of what

I'm going to tell you right now you're just going to write down as notes but they're very important notes first thing this is what the Harmonic series is defined as the harmonic series would be this if you look at the series if you start plugging in one and then two and then three and then four you're going to get 1 plus a half plus a thir plus a fourth plus forever and ever and ever 1/2 1/3 1/4th 1/ fth 1 6th 1 7th 1/ eth 1 nth 1 10th and so on and so on and

so on and so on and so on forever and ever and ever and ever you guys okay with that one now you would think because this is Positive and because it's decreasing because it's doing that that you would get a conver series you do not the harmonic series is Divergent I won't prove it to you the proof is in any calculus B that you find it's not that hard I think you can follow it the harmonic series is actually Divergent oh this series diverges now there's a couple things we learn out of doing the proof

on this Thing I'm going to give to you first one is this they very interesting Concepts I I I want you to know first one is this um if a sequence going back to sequences for a second if a sequence a subn is convergent any subsequence of a subn must also be convergent does that make sense to you I say hey here's this gigantic sequence This gigantic sequence converges we know that if I cut a little bit off of it and say now let's just look at the inside part of it can this possibly diverge

when the whole thing converges that doesn't make sense at all so if the sequence converges any subsection subsequence of it must also be convergent that wouldn't make sense otherwise you can't have that now the Corel of this is this sorry two Sh what that means is the Cory think about it this way let's say that you had a gigantic series that you didn't listen carefully please that you didn't know was convergent or Divergent let's say you didn't know it's the whole thing okay now let's say you consider just a subsequence of that of that sequence

if you find out that that subsequence is Divergent what's it tell you about the whole sequence has to there's no way for a little part to be Divergent and then Come back oh but the rest is convergent that doesn't make sense okay so if a subsequence if a part of your your sequence if a subsequence is Divergent the whole sequence is Divergent so to show Divergence to show that a sequence diverges it is enough to show that a subsequence diverges that's actually what they do with the harmonic series on the harmonic Series you take a

subsequence you show that subsequence is Divergent to Infinity but just the subsequence of it uh it is Divergent therefore the whole sequence diverges do you guys understand these two statements hey if a sequence is convergent every subsequence has to be as well therefore if a subsequence is Divergent the whole thing has to be Divergent so sequence converges all sub subsequences have to as well but if a Subsequence diverges the whole sequence must be Divergent as well couple more notes for you actually you know I still got some room so more notes you know what I'm

going to write over here cuz really never mind the reason I give you this one about sequences is because we actually used that for the partial sums of the harmonic series it's what the way we proved that it's Divergent U we're going To go back to Series right now as for some more notes um these these stem from a lot of these ideas if a series converges if a series converges then the limit the limit of the sequence of terms has to equal zero what this means I hope that you you grasp on to this

uh please listen carefully you remember talking about sequences and how when you take the limit of a sequence if the limit of a Sequence equals a number then it converges and if it doesn't equal a number then it diverges remember that well think about this now it's not just that we have the ending terms of our sequence being numbers it's that we're adding them together so if you're see listen carefully because I don't want to blow your heads up okay um if your sequence if the limit of your sequence goes to a specific number that's

not zero do you Understand that you're always going to be adding more of that number so if your the limit of your sequence ends in like five oh good great your sequence converges but then for Infinity you're going to be adding 5 + 5 + 5 + 5 + 5 + 5 plus 5 what's the only number that you can keep on adding and it doesn't add to your sum Z that's what that says it says that in order for your series to be convergent in order for this to happen if if this happens if

your series is Convergent then what must have taken place was that this the limit of your sequence of these terms eventually has to go to zero you have to end by adding terms that are zero show pant if you understand that concept that's the idea here now what that means is that so to show a series diverges it's enough To show that the limit of your sequence of terms does not equal zero if the limit of your sequence of terms from which you are getting your series does not equal zero or does not exist the

series from which we're the series upon uh the series that we're basing Upon Our sequence of as subn terms Dees here's the problem that students Have with this here's the problem the students have problem students have is they think that this is a bond conditional statement they think that if the series converges the limit equals zero and if the limit equals z the series converges that is not true I'm going to say it right out flat out that is not true the times it happens sometimes it doesn't here's what I'm going to tell you I'm

going to go through it one more time so I'm trying To be very clear here with you if you know I'm going to speak louder so you really hear me doesn't mean you're going to understand it more but you're going to hear me if if the series converges the limit of your sequence from which you're getting your terms for for your Series has to be zero that has to happen because you're adding terms which are going to 0 0 0 it is not true that if the limit of your terms equals Zero of your sequence

that the series is going converge prime example right here the limit of 1 n is z does the series converge no the series diverges and it can be proven I didn't prove it but it can be proven so if we want to show convergent we don't just do this we don't but if we want to show Divergence you go okay well cool if it converges this has to happen so if this doesn't happen it will not converge does that make sense so we can Do that though we can say cool so it's enough to to

show a a series divergers it's enough to show this happens so if the limit of our terms does not equal zero then what that means is we're not adding zero at the very end of our sequence we're not adding them together we're adding numbers that goes forever and ever we're gonna always add up numbers more and more and more and more and more that are the same okay or close to the same or it doesn't exist then This Series has to diverge that that's what we can do um let me say that one more time

so big note limit equals z does not not necessarily I think I spelled that wrong mean convergence of your series it does not necessarily mean convergence of your series you know I was going to say one More thing I completely forgot what it was than totally for that's important too looks like we can't do our yeah too bad it'll come to me sometime okay so if the series converges this has to happen if this happens it does not necessarily mean your series converges but if this if you want to show that your series diverges just

show that this doesn't happen does that make sense to you this like a condition you got to have this And something else for it to be convergent so this doesn't happen bam your Series diverges this right here I know what it was this is called the Divergence test so if you want if they're asking you hey do the Divergence test all you got to do take the limit of your terms please listen carefully if the limit of the sequence from which you were getting your series if you the limit of your sequence does not go

to zero or does not exist what do you know About your series if the limit of the terms of your sequence do go to zero what do you know about about your your series it might not diverge Michael so this is due Thursday okay um there's one more thing we got to do it's super super easy uh let me show you an example of this first I'll show you one more thing we can do and then uh that's a w well you got to you got to see this One time okay at least this so

we're going to use the Divergence test real quick and when I mean real quick it's going to be real quick here's your series here's how you show whether it's divergent or maybe not you take the limit as your terms approach Infinity of the terms of your series that's a sequence the terms of your series is a sequence so you take basically the limit Of your sequence how much is this limit 2/3 it's 2/3 that's what it is because you divide by the larg power L rule twice whatever it is simplify you get 2/3 is this

equal to zero what do you know about that series here's why because at the very end when you get towards Infinity you're going to be adding fractions together that are of the value 2/3 2/3 plus 2/3 that eventually will still go to Infinity does that make sense to you That's the whole idea we have one more thing to do it's about five minutes I'm not going to do it now don't worry uh but next time we have a little bit more to do on this section so the last little bit section 9 point whatever we're

on uh we're going to talk about properties of convergent series like I said we're going to do just maybe one or two more examples so the first thing I need you to know is that if we know that our series is Convergent or so basically if we're testing for our convergence what we can do with it are these two properties if we ever have a series with a constant being multiplied by the sequence from where we get our Series so like our H subn we can always pull that constant out in front of our Series

so instead of having series of C * a subn it's the same thing as doing C * the series of a sub with no problems also what you need To know is that if you have a series where you have two sequences added together that are giving you the terms of the series or subtracted so added or subtracted you can split up those series by the sum or the difference so if we have a series that's of two sequences added together we can have a series of the first sequence plus a series of the second

sequence that's the idea you guys okay with that one and often times this Will really help you on determining whether or not your series are convergent or Divergent and if they are convergent it'll help you to find the sum so let me show you with this one final example and then I'll show you that shortcut I was talking about so first thing do you see anything that fits our two properties that we have what could we do here Theus yeah we could in fact if you think about this we can separate the fraction We could

have 2 the N over 3 the N minus 5 the N over 3 to the n and then separate those series does that make sense to you yeah so we could do this and say all right well cool this is uh and = 1 to Infinity of 2 N over 3 N - 5 n over 3 n that works we can always separate fractions like that and then because I have one sequence minus another sequence inside of my series here I can separate those that series Now now that we've separated our series we can actually

start looking at each one and seeing whether they're convergent or not let me make one statement about this please before you keep going on this listen carefully our original series was this one correct so if we split it into Parts if we split it into these parts I don't I don't care how many there are if we split it into Parts tell me what has to happen with each part to make sure that this whole Thing is convergent tell me what has to happen all of them say that louder all of them have to be

convergent very good each part must be convergent for the whole thing to be convergent does that make sense to you you can't have one thing add to infinity and the other thing add to a number and go oh therefore this thing adds to a number that doesn't make sense so if any one of these parts is Divergent the whole thing is Divergent does that make sense to you Yes what that means is that if you're smart about this and you see one part that's definitely Divergent boo yeah you're done does that make sense yeah I

mean I mean why would you go through the process of finding the sum of one thing if you know the other things can be diverg Rob question suppose they're both convergent will they arrive at the same number these both these won't be they won't be the same number but you could add them together to get this series That's what this whole thing says okay it says uh the series of a sum is the sum of two series the sum is the sum suum will be the same if you add them together if they were both

convergent does that make sense to you so here we go if we did this one thing that we could do is we could make this a series N = 1 to Infinity of 23 to the nus the series of n = 1 to Infinity of 5/3 to the N can you explain to me why I'm putting it in this format geometric That's a geometric Series yeah if we have a geometric Series right now we can tell immediately whether these things are convergent or Divergent um now there's one thing that's wrong with this picture do you

see what's wrong with this picture as far as my series are [Music] concerned no one of us does I know one of us does one of us SE I see what's wrong two of us then cuz one other person said I know I know what wrong Maybe they're a lying I don't know um but let's see if you know what's what's incorrect uh I told you that with geometric series in order to find the sum correctly your powers have to match up your index and your powers have to match up it's got listen in order

for the sum formula to work remember the sum formula was a over 1 - R do you remember that at all yeah in order for that to work your first power must be zero is your first Power zero no okay so This formula won't work correctly now I'm about to give you a shortcut in about 5 minutes on how to get around this all right without doing a whole bunch of work there's two ways really but even if it doesn't match up exactly you can still tell me whether it's convergent or not because this R

will not change even though I strip off a fraction and make my my power go down one so here's my question let's suppose my R is 2/3 would this be a convergent Series yes or no 2/3 let's say this is 5/3 would this be a converged series yes no no so automatically we know this is Divergent now let's assume that they're both converging here's what you would do to get my powers to match up I would split off one power and then that reduces my exponent by one does that make sense to you you sure

this is the same thing as that but now they match hey hey now it's a geometric series The Way We like to see It where our powers match and if your powers match up basically match up means when I plug in my first number my first uh n when I plug that in this power is zero please listen very carefully here if the first power is zero then a is actually the first term in my series do you understand that yeah this is always the first term in your series if these match up are you

clear okay so this would be the geometric series your a is 2/3 your R is 2/3 that would be your sum it'd be 2/3 over 1 - 2/3 so hands feel okay with that one now here's my next question should I have done that no let's go to the next one minus tell me something about this series what's my what's my R again so you could you could strip this off and do n = 1 1 to Infinity of 5/3 * 5/3 nus 1 you could do that but notice when you strip off one of

those fractions does your R change at all just the power change so your R is Not going to ever change here tell me something about the r is it more than one or less than one greater than bam you know something about geometric series when the r is greater than one tell me about that so we got a sum minus something that is Divergent if you have a sum minus something that is Divergent this is a real number right that's something this is I don't know what that is it's Infinity actually Because you have 5/3

that's getting bigger and bigger and bigger because it's more than one this is going to be Infinity so you have something minus infinity can you tell me what the sum of my series is if you have something minus infinity no if listen if you break up a series each one of these parts would have to be convergent for the whole series to be convergent if one little piece is Divergent then the whole series is Divergent n you just heard me feel Okay with that so let's Imagine This was um 13 and this was 1/3 would

this be converted in this case you would do 1/3 over 1 - 1/3 You' figure out this sum you'd figure out this sum and you would subtract them and that would give you the sum of your series show P if you okay with this idea now do you want to see a little shortcut when you deal with this stuff like this where the powers don't match exactly you Want to see a shortcut for that yes okay sometimes I'll teach it sometimes I don't I think you guys can handle it so I'll give you I'll give

you the shortcut okay so here's our formula if we have this or we have this if we have this or this well then we know that the sum would be given by those two expressions it doesn't change even Though my index is different even though my power is different here's the whole reason why this doesn't change whole reason why this doesn't change is because I don't care what this is if my first power is zero that means I'm going to get a as my first term does that make sense to you it's always going to

be the first term if my first power is oh 1 - one that's zero my first power is zero a is going to be my first term so basically basically listen carefully Here because where I'm going to get my shortcut from basically a stands for the first term in your series look at this if I plug in zero for my n what's R the n one a would be my first term in my series you can make shortcuts with this let me give you an example I'll give you one like this one tell you what

instead of actually Matching this up here's what you can do with it if you understand if you understand that a is always the first term in my series then here's what I can do I can plug in one here um what's my when I plug in one what's 2/3 to the first Power I can take my first term in my Ser notice how 2/3 would be my first term in my series if I actually do this right I plug in one then I plug in two then plug in three 2/3 would be the very first

one wouldn't It 2/3 over 1 - 2/3 you just take the first term you actually do the do the work plug in whatever this index is take the first term one minus whatever your R is because we know our R is not going to change by stripping off these fractions your R don't worry about your R it's never going to change it's as soon as you get in this format your R would never change you're just worried about your first term since I've shown you here that a is always your first term we Can kind

of cheat at it a bit did that make sense to you yeah I'll give you one more example so that way you can you can really see it okay this one's uh I think I just made this one up a little while ago and I want you want to show you so example you guys ready for it I'm going to do much with this example so stick with me of course before we do anything with this you need to recognize that this is a geometric series and the way that I See that I go well

wait a minute I've got a constant easy constant to a power and I have a constant to a power that's based in N that right there says to me geometric does that make sense to you it's it's hey it's a constant to an to n and a constant to an N that means if it's not the same n i can strip some off or put some in whatever uh but ultimately I'm going to be able to get a fraction to an N power that's what says geometric about this to me Now we can work with

this it's not too bad if I have n = 2 to Infinity of e n over 3 n + 1 that's the same thing as 3 * 3 n show if makes if that makes sense to you now I'm going to break this up a little bit I'm going to get n = 2 to the Infinity of 13 notice my 1/3 and then I'll have e over 3 to the N I need a big show pans if you Understand the algebra of this thing in the middle we okay okay now here's my point this is

a geometric series correct correct yes here's my question do you know how much the a is yes if you tell me 1/3 you're wrong is the a 1/3 no no the a is not 1/3 here's why the a is not 1/3 does your index match your power which means when I plug my index in is my first Power zero that's how geometric series Works okay if the first power is not zero that means this is not your first term that's the problem does that make sense to you that's the problem this has to listen I'll

make it very cut and dry for these formulas to work this has to be your first term if it ain't your first term that formula don't work does that make sense so the reason why I I rarely teach I think you guys are smart enough you'll pick it up okay the reason why I rarely teach this is because People when they get this they forget that after the match and they do this they go oh my a is 1/3 I can do 1us e over3 big mistake big mistake there's now two options that you have

for going around this the first option is the old way and it's not that hard all you got to do is just match up this this with this so if this is two if this is zero I need n if this is one I need n minus one if this is two I need n minus that's it so option number One you could do it the old way and just do okay if this is starting at two I can do 1/3 I can split off an e e over 3 S so basically e^2 over 3^2

I can split that off split off an e over 3 2 and then I'll have an e 3 nus 2 I'm curious do you see that this is the same thing as this do you see that yes do you see that this whole thing's a constant yes that would be n = 2 oops that's not right 2 to Infinity this would be e 2 Over 27 verify that for me please this would be e^2 over 27 now let me ask a few questions about this if you plug in two is your first Power zero yes

that means this is geometric and it fits my form all you have to have is the first power being zero because then what that means is this is your very first term if this is your very first term then I get to do my sum formula as e^2 over 27 1us e 3r notice how no matter what Our e e over3 sorry not e 3r e over3 our e over3 is not changing even though we're stripping off parts that fraction it's not changing that's now my sum yeah of course you would simplify that right but

I'm just trying to give you an example here did that one make sense to you now here's a shortcut for doing this here's a shortcut check it out from right here so or not both or if you're good at this if you're really good at this here's what you do You understand that what a is is really the first term in your series you go okay well duh if this gives me 2us 2 is zero this whole thing is one that's the first term I'd start with well you can do that from here just plug

in two if you plug in two you're going to get look at that e s 9 huh e s do you see the E squ 9 I'm looking do you see the E Square 9 if you got e^2 9 e^2 9 * 1/3 is e^2 27 hey that's the first term in my series is the r going to change No 1 - e Over3 do we get the same thing it's just a little bit less steps the only problem and I'm I'm praying that you see this students have an issue when they don't understand what's

going on they don't understand this is the first term that's what a really stands for that that's the first term of your series and what they end up doing is go oh my a is 1/3 Mr Leonard says I don't care if those match up no Mr Leonard says care you got to Care you have to this has to be an N minus 2 for you to legitimately use that formula it's got to match up your power with your index it must if it doesn't then you have to think harder you think okay well I

know that a is really just my first term of my Series so if I plug in my first index of two the first term my series is not 1/3 it's e 27 that's the idea show that makes sense to you cool excellent job also one last bit of advice do not forget about the Divergence test don't forget about that you can check series very quickly by Divergence test and that's what you should do every time you get a series take a limit of it take a limit if that limit does not equal zero what do

you know about your series Divergence that'll save you a lot of time on this homework okay now the next thing we're going to talk about questions so pretty much what you're Doing is whatever n equals you just take it to that power if this is z minus 0 if this is 1 - one if this is 2 - 2 if this is 3 - 3 if this is four - 4 you need the first power to be zero if that's true then a stands for your first term if that's not true you figure out your

first term make sense yeah yeah now