 to be less and less people, I guess it's because they're all watching the winder. So, I haven't had good luck, as you may have noticed with the clickers and everything. So, I'm just, do we give up on it? So, I'm sorry. So, I spoke to the other instructor, she's having exactly the same problem as I am. It's just not working. Sorry. So, you don't need to bring your clicker anymore because it's not working. Okay, so, what we were talking about and what I want to talk about briefly is this idea of an infinite sequence, which is not the main point of this section of the class, but it's a tool that we can use to understand infinite sums. Back where you belong. Sorry. Okay, so we're talking about infinite sequences, which is just, this is an infinite list of numbers. And either this list converges or it doesn't. So, it's something like one minus a half, a third, plus minus a quarter, two definitely minus. Let's make this minus. Minus one, minus a third, quarter, minus a fifth, six, et cetera. And in this case, if you wait long enough, this gets as close to zero as you like and it stays there. So, this converges to zero because, so this converges to zero. Now, one thing that you need to get used to doing, which is not hard in this case, is instead of writing it as a list like this, write a formula for the 17th term or the 93rd term or in general the nth term. So, usually we can call this the first term, the second term, the third term, et cetera. And you can figure out a formula here for the nth term. So, what is the nth term? If you look at this, it's pretty obvious, or should be pretty obvious, that if I go to the tenth term, it will be a positive number because the bottom is even. And it will be positive one tenth. So, if I look at the 103rd term, it will be minus one over 103 and so on. And then showing that this goes to zero now comes down to showing that for n large, this number is this goes to zero as you like. One to the n, this means minus zero, not a function. So, I can't really say f of x is minus one to the x over x and claim that the limit as x goes to infinity of f of x to zero because this doesn't mean anything. I have no idea what minus one to the x means when x is not a whole number. Well, okay, I guess I do kind of know what it means if x is a rational number, but I have no idea what minus one is a positive number. So, this sort of doesn't make sense. Of course, we can just look at the absolute value and then notice that the limit as x goes to infinity of one over x to zero. And since the absolute value goes to zero, this goes to zero. This is sort of not a good argument because this doesn't make sense. So, this is one subtlety about sequences versus functions, but it sort of behaves kind of like functions. Another similar kind of thing. Suppose I want to look at, say I want instead of that, I want, well, I don't know, you could play with those in the whole list. So, let me also point out, say I want to look at the sequence n over n factorial. This is a little harder to think about whether this converges or not. Does it? Some people say, yes, no. So, I hear some gas, and I hear no. So, we have those two choices. And I guess the third, I don't know too hard if we can figure out. And that's another choice. So, you say diverse is y. So, let's make that a little more real. This is your right. Let's make it a little more real. So, a1 is 1 to the 1 over 1 factorial is 1. a2 is 2 square over 2 factorial. a3 is 3 square over 3 factorial, which is 3q over 3 factorial. Which is 3 times 3 times 3 over 3 times 2 times 1. To the fourth is 4 times 4 times 4 times 4 over 4 times 3 times 2 times 1. And so, in general, this n over n times n over n minus 1 times n over n minus 2. Oops, yes. 1 down to n over 1. And if we look at this, this is always bigger than n. Because this is 1. This is bigger than 1. This is bigger than 1. This is bigger than 1. This is bigger than 1. And we have n and n. So, this is certainly bigger than, well, this equals 1. This is bigger than 1. So, I can replace it by a1 and make something smaller. This is bigger than 1. So, I can replace it by 1 and make something smaller. The next term is bigger than 1. So, I can replace it by 1 and make something smaller. And at the very end, I'm going to leave the n over 1 over 1. So, the n term is always bigger than n. And we know that the sequence 1, 2, 3, 4, 5 diverges. Add 1 every time. When you look at it, it might look like it's going to converge. But in fact, it doesn't. You have to analyze a little bit to see that it won't. You can't just say, gee, it seems like they're getting bigger. Similarly, to see whether something converges, you can't just say, gee, it seems like they're getting smaller. You can't just look a little harder to show that it really is getting bigger or really is getting smaller. So, I mentioned before that this comparison test that we looked at for integrals will come back. Here it's coming back in the context of sequences. I'm comparing a complicated sequence to a simple sequence. Every term in this simple sequence gets really big. Or the terms get bigger and bigger and bigger. And this complicated sequence gets bigger faster. So I have a comparison here that something that gets really big pushes up something complicated. So the complicated thing cannot converge. The same thing that we did when we looked at improper integrals and we compared them to more simple integrals. If the integral diverges and we have something bigger than it and that diverges too. Same idea. Okay, so a lot of times to attack these things, one way to do it is to group things intelligently and then make them smaller or make them bigger. Notice that if instead of doing n to the n over n factorial I could apply a similar argument which I won't do unless somebody really wants. You didn't even know what I'm going to do. Okay, let's prove Sard's theorem. That's what I'm doing in my topology class. You want to do that? It only takes three classes and it's just part of that. But it's okay. Okay, so similarly you should be able to see by the same argument that this converges. Because if we just flip everything then instead of making things smaller we can make things bigger and we can get a 1 over n. So we can see that this thing is smaller than so I won't write out the details but n factorial over n to the n is less than 1 over n and 1 over n goes to 0. So n factorial over n goes to 0. It's kind of like another thing that may have confused you in calculus. The differential calculus is a squeezed theorem. A squeezed theorem is really a comparison theorem. It's the same kind of thing. Right here we have just by a picture 1 over n does this and this sequence, well, they match up at 1 but then the next term A2 is 2 over 2, 2 over 4 which is a half. Well, I guess they still match, right? A3 is, wait a minute. They do 1. Okay, yes. 6 over 9 which is a third. Why are they equal? What am I doing wrong here? That's not 9, that's 27. Which is less than a third and A4 is 24 over 64 which is certainly less than a quarter. So this guy gets pushed down here and so since this one goes down, that is drag. 4 to the fourth is not 64. So you can squish them in there. The best way to figure these things out is to play with quite a few of them so of course I gave you a bunch on homework, yeah. Another content. So I mentioned at the end of the last class we can also define these sequences recursively. This comes up a lot in certain courses of study like computer science but also in other things. So this notion of a recursive person definition would be something like, I tell you, well, so one really well known one is say the Fibonacci sequence. This one doesn't converge but it's well known. So how many people know what the Fibonacci sequence is? A couple. How many people know what the Fibonacci sequence is? Okay. So the Fibonacci sequence, we take the first term to be one, the second term to be one and then from here on we just add the last two together to get the previous one. So a3 is one plus one, which is two, a4 is two plus one. Maybe my numbers are off, sometimes people start at zero and so on. So a5, a5 is now this plus this and in general, an is an minus one plus an minus two. The Fibonacci came up with this as a way to, in his thought experiment where he came up with this sequence, he was describing rabbits. So you have a pair of rabbits. They're too young to breed so they don't breed. The next one takes rabbits to go up. The next month you still have a pair of rabbits. But then the next month they breed and now you have two pairs of rabbits and then the next month these two pairs breed and you have three pairs and then the next month the three pairs breed in zone where you get this sequence and this grows really fast. It doesn't look like it yet but this goes really fast. So for example, a6 is five plus three is eight, a7 is eight plus five is 13 and you start to see this grows exponentially fast. So the idea of recursive is that we define terms of previous A again. So it's possible to write a formula for what the 15th term of the Fibonacci sequence is. I won't do it. It has to do with the golden mean. But anyway, so it has to do with square roots and stuff. But sometimes it's not. But this gives us another way to define sequences. Now, I'm going to put aside the notion of sequences. I'll give you, you're going to do an x-paper homework or we'll also do some of this kind of stuff. But I want to talk about something else which is a special kind of sequence which unfortunately goes by the name of series and because it English the word sequence and series means pretty much the same thing, it gets confusing. So I'm going to try not to use the word series instead of going to use the word sum which does not mean the same thing as sequence. I'm going to talk about infinite sums now. I'll put series here in parentheses because in the book they're going to be referring to the series and you'll often see that they counter the series but I will always try to say sums. These are special sequences. So this is a special sequence where the sequence of numbers that I get I get by adding stuff to what I already have. So for example, let's take, and I'm going to use s now to indicate that it's a sum. So I'm going to make the sequence s1, s2, s3, etc. So instead of calling it a, I'm going to call it s to emphasize it's a sum. And let's take s1 is equal to 1 and then s2 to that I'm going to add one half squared. And then s3 I'll take whatever I had before so it's sort of a recursive thing plus 1 third squared. And then s4 so this is 1 plus 1 half squared plus 1 third squared. Actually this isn't what I wanted to do. Let me do one that I can actually add up. I can add this one too but let's not do it now. Well okay so let's do this one for an example and then we'll find out. And in general sn will be the previous guy plus 1 over n squared. So I just keep adding a little bit each time which in this case 1 plus 1 half squared plus 1 third squared plus 1 quarter squared plus 1 over n squared and then I stop. And as I said as we've seen before we have a nicer notation for this this gives a little tedious to write out so we have another notation that we can use that scared some people at the start of the class so I'm going to start at n equals let's use instead of n let's use i equals 1 and then I'm stopping at n of 1 over high squared. So this sigma notation just means add these things and I tell you where to start and where to stop. These two notations are exactly the same thing I used it at the beginning of the class several people were confused several people had no problem with it but we're going to use it quite a lot from now on so if it bothers you we need to get used to it. So this sigma is a Greek letter capital letter s it's supposed to make you think of the word sum so it's supposed to make you think of adding things. Is everybody clear on this notation? And for those of you that dropped down from 127 hey it's like the first day again. Okay. So I'm not going to say whether this makes any sense well this makes sense but what we don't yet know is how to make sense of whether this is a number or not as n gets large as n goes to infinity does this have a limit or not. And that's one of the central points of what we're going to do for the next while in this class. So in general so the name for this s n is called the sequence of partial sums in other words to n terms and then I stop and I see what I got. And the question is if I do this and I stop it in terms well instead of writing one over i squared let's just write a sub i does this limit as I let it go to infinity so this is the same thing as the limit as n goes to infinity with s n does this converge or you can't say because it's just too hard it either does converge or it doesn't but sometimes it's hard to know now in this example I'm using one over i squared I don't want to quite do that but yet it started easier once but this is sort of one of the central questions that we'll look at for a while given a setup like this does this converge I remember what I said last time a real goal is not to do this just for lists of numbers but to do this for lists of functions we want to add up we want to make infinite degree polynomials and see if they make sense we want to add up things here where there's an x involved but we'll get there and that will give us a new kind of function so is this where we're going made sense ok so so let's work on some examples and see this is why we need to introduce at least a little bit about sequences because often until we build up some tools in order to show that a series converges or a sum converges we turn it into a sequence so let's start with something relatively easy that you've probably all seen before you've probably saw it you might have seen it in middle school I think it's in the middle school curriculum suppose instead of letting my numbers grow and I fix the number so I want to look at this where I say take a half and raise it to get power and so I want to know does this make sense this means I'm looking at the c and let's start with zero instead of 1 so one of the things that you have to pay attention to is all of these symbols here mean something we're starting here at zero a first term we go to infinity and we have some jump here and that fix it so this means we want to look at the sequence the first partial sum of one half to the zero is one our second partial sum is one plus and now we let n be one one plus one half our next term in the series is S3 which is one plus a half plus a half square which is one plus a half plus a quarter which is something I don't know I'm not even writing down and so on and in general as in one plus a half plus a half squared half cubed and then we stop at one over two to the n now this has a chance of converging because I start with something I add something smaller to it and to that I add something half that size to that I add something half that size so I start with something one unit tall make it one unit tall actually this is area one let me give you a geometric term for this I'll prove it now that it's being said I start with something of area one and now I want to add something of area one half so let's make it one unit wide but only half is tall so this has area half and now I want to add in something of area one quarter so instead of being half let's make it I'm going to be half a unit tall by half a unit wide and now the next term is one eighth to get something of area one eighth well let's take half of a quarter and that's a sixteenth what do I do wrong here one and eighth it's quarter of a quarter anyway I can just add in these little bits here they should add up to two I'll just continue this pattern and I see that it fits into two so that's sort of an idea that this should add up to two but let's make it a little more precise how many of you have seen this before how many of you not seen this before you're wide how many of you have never seen a geometric series before see there's less people some of you are liars not liars so how can we do something obviously I'm using a two here but any number this adds up to without drawing little pictures so if I look at that list one plus a half plus blah blah blah look at Sn it's a list what do I get if I take half of it so Sn is one plus a half plus a half square plus et cetera up to one over two to the n suppose I took half of Sn what do I get I'm going to start over here I get a half plus one is a half one half times a half should I write people understand where these terms are coming from anybody confused you're confused what would you just find the value of the sum of Sn and then multiply that number by one half the one half I don't know what the sum of Sn is well I mean I do but if I haven't figured it out I don't know what it is right so if n is one million and you don't know the formula for a partial sum of a geometric series how are you going to figure it out you're not just going to punch it into your calculator how are you going to figure it out you can't so I have to be clever and figure out the formula for Sn so the way I'm going to figure out the formula for Sn let me put it in two steps is by being clever and noticing if I take a half of that it gives me the same thing to shift it over by one because a half of one is a half a half of a half is one over two squared a half of one over two squared is one over two cubed so a half of each term that I get here just gives me the next term so I'll pick up this one from the two to the n minus one and then I have a half of this and now I can be clever and subtract so if I subtract which way do I want to subtract it doesn't really matter but let's subtract this let's do this minus this well if I subtract here well Sn is some number and Sn minus a half of an Sn is still a half of an Sn but here when I subtract Sn minus zero is one and half minus half zero is zero zero is a good minus let me just write it here one over two to the n plus one so that means that Sn is twice this I figured out what Sn is I have a formula that tells me so I have now a term to this sum which is nasty I have to add a lot of things into a sequence so if I add up the first 53 terms I will get two times one minus one over two to the 53 which is very close to two and if I add up the first million terms I get two times one minus one over two to the million and one so I know now that so I know now that the n of one over two to the i is two times one minus one over two to the i plus one if I add up n terms oops not i if I add up n terms I get this so now I can take a limit because I have turned it into a sequence and as n goes to infinity this goes to zero and adds up to two just like in my pictures there was nothing magic about two here and so let's do it again quickly for instead of two the conventional letter is r so if I do the same game with r so I'm not going to write out all the steps I'm just going to point so if I have n equals zero to infinity let's just call it r of r to the n it's a number between minus one and one so it's a fraction and then I can play the same game I guess I'll write a couple of terms one plus r r squared r to the n if I multiply it by r I get r r squared r to the n then I can subtract stuff cancels cancel, cancel, cancel and then I can divide so this works for any r but if r is less than one and absolute value this will go to zero when I take the limit so I get one over one minus r I didn't I multiplied by r and then I subtracted so I subtracted and then I killed, killed, killed, killed and you left with just two numbers instead of a huge list so the goal was to get rid of this jump it had to be a little clever for knowing how to do it to figure out how to do it but that's what I did so what I've just shown is that well two things if you want to add up a bunch of powers the sum of a bunch of powers is that so if I want to add up one so I want to add up one plus two squared plus two cubed plus two to the third plus two to the fourth plus up to two to the hundredth the answer will be two to the hundredth first plus one divided by one minus yes this is a minus yes paper because I subtracted so when I subtract this is minus thank you so when I did the example of two I was wondering why I wasn't working good so anyway but we've also shown that if we add up a bunch of powers infinitely a lot this is one over one minus r if r is not too big in absolute value now realize of course this tells you something else so this tells us that if I want to add up the powers of a third that I get one over one minus a third that is three halves but really this is also a function I want to emphasize that this is telling us more so this is called a geometric series unfortunately nobody calls it a geometric sum but it's a geometric series and it's because it arose in a geometric situation where you're chopping something like this but notice that in addition to this something if I look at this same formula in a different way I mean it doesn't matter what letter I use if I use r or x or a banana it's still true so I've just shown that one over one minus x is one plus x plus x square plus x cubed plus x to the fourth blah blah blah forever so this has sort of given us a different way to do division in some sense we can do division by adding and multiplying as long as the number is small we'll come back to this but here we have sort of an infinite degree polynomial and this infinite degree polynomial is this rational function because it doesn't matter whether I use r or x okay so there's one example of an infinite sum and even though it seems weird to be adding up infinitely many things it makes some amount it makes sense you can make sense of it as long as those things shrink fast enough but it is not sufficient for those things to just shrink to zero if they shrink to zero too slowly so then they won't add up I mean they will add up they'll add up to too much stuff so let's look at another version of this kind of thing notice that the limit as n goes to infinity of one over x is certainly zero well no it isn't it's certainly one over x one over n is certainly zero that is the sequence one one-half, one-third, one-quarter one-fifth, blah blah blah goes to zero what about let's start with one if I add up the terms so this is one plus a half plus a third plus a quarter like that it seems like this should converge but it doesn't this does not converge this is called the harmonic here if you know this name but if you hear the name then you know it and this does not converge and let me show you why it doesn't converge it seems like it should and in fact if you get out your computer and you start calculating it will seem to converge but that's because your computer has finite precision it converges it diverges incredibly slowly so I can get this as big as I like but it takes a really long time to get bigger than ten say let's go through that so let's just start looking at the partial sums and see what we can do so s one let me just write out s n for n large so let's just write and one plus a half plus a third plus a quarter plus a fifth plus a sixth plus a seventh maybe that's enough maybe I'll need to add some more and now what I want to do is I want to start grouping things and I always forget where I want to group them so I want to compare this to I want to group them into things that are bigger than a half so I'll just start with the one and I'll leave it alone and a half is just fine so a half is fine here but notice that one third is bigger than a quarter so this is bigger than I leave the one alone let's leave the half alone a third is bigger than a quarter so if I replace the third with a quarter no problem and I'll leave the quarter alone and now a fifth is bigger than an eighth and so is a sixth and so is a seventh so I'm going to replace these guys with quarters and I'm going to replace these guys with eighths a ninth, a tenth an eleventh a twelfth I don't want to go all the way to sixteen I'm going to replace the next set and in general I'm going to replace these with powers of two that add up to a half a ninth through a sixteenth each of those terms is less than a sixteenth now why did this do me any good well now let's add one plus a half and then if I add these two together I get another half and then if I add these four eighths together I get another half and then if I add these eight sixteens together I get another half and then if I add the sixteen thirty seconds together I get another half and then if I add the following thirty two sixty fourths together I get another half so this is bigger than one plus as many halves as you want so this goes to infinity I take off things and made them smaller and I wound up with something that gets as big as I like it takes a long time to get there but it gets as big as I like so again I use this comparison test to show that the harmonic series is bigger than the series I'm just adding a lot of powers together I just have to wait a while to add together those things so I will pick up with this kind of thing do you want to see it? yeah