 So let's remember what we're doing, so we're looking at convergence, we're divergence of infinite series, that's what we're looking at. And right now we have a few ways to do this and the goal for today is to build up some more. So I'm just reminding you what we know. So they're sort of just mess around. So just take the thing, adjust it, see what you can see happen. That was the first technique we had. So like for example, if it's a telescopic or telescopic series, maybe partial fractions will help. If you can add up, let's say this better. So we somehow calculate partial sums, SN, which are the sum from wherever you start, I better not use N, I equals 1 to N, the AI, and then look at the limit. I'm going to say N goes to infinity. So the sort of ad hoc, whatever you can make work kinds of techniques, it'll be more useful. See if it's a geometric series. We know that the sum N's power adds up to 1 over 1 minus R if R is less than 1, and it's the value. And then we have some tests that we developed. So if the terms are all positive and decreasing, if it's smaller, we could use the integral test. And the integral test says that if you have some function F of X which matches, I guess F of N, if F of N matches with the N's term, then the integral from 1 to infinity of F of X DX converges or diverges. So let's just say it does the same as the sum. We also know that, yeah, yeah, like I said, I have trouble with these things. I said the right words, I just wrote the wrong words. I wrote the wrong symbols. More other ones too. This one's right, because this one I realized as I was writing it, it was wrong when I fixed it. These can be equal. So it's decreasing series of positive terms than N. We also have, again, well, so using the integral test, we determined that a special class of series called P series, which is 1 over N to a power P. So it's the P that, well, 1 over N to the P. So just to emphasize what this means, this is 1 plus 1 over 2 to the P plus 1 over 3 to the P, blah, blah, blah, because this converges if P is bigger than 1. So if we have the sum of powers, a fixed power of 1 over N, then this converges for big powers and diverges for P less than 1. P doesn't have to be a number. It can be, for example, a square root. So this is also what we've done so far. And then since we have at least two model series that we know about, we can compare things. So again, if the terms are positive, that's the right way to write positive, I think, yeah, non-negative terms and also BN are non-negative terms, then we can compare, so now we have two situations. If the ANs are bigger than the BNs, Ns blow up and so do the ANs. So we've got, I'm not sure, I guess it's the same answer than that, if we have a, so this time I made the series I'm comparing to small, if I have a small series that blows up, then so is the big series. And the reverse, if, let's keep it this way, the ANs is big, well, now let me flip it around. If the ANs are smaller than the BNs and the BNs converge, so does, so notice I reversed the sense of the comparison. Someone please check that I got my inequality right the right way. I do, or I don't. Oh, you have a question? It's good? Okay, yes, good question. No, no, no, no, no, well, okay, you're saying if this is big, so if AN is bigger than BN and I know BN diverges, then I know about AN, smaller than BN, so I need to pick BN, so the point of comparison is I have to think I want to know about, if I believe that that thing blows up, then I look for something smaller that also blows up, if I believe that that thing converges, then I look for something bigger that also converges. If I can't find anything, well, then I didn't learn anything because I couldn't find anything to compare to, right? This is sort of, I don't know, guilt or honesty by association, so if you want to prove that somebody is guilty by association, you've got to find guilty friends for them to hang around with, and you want to say, he's a guilty guy because all of his friends are honest. This doesn't convince any of you, right? So, okay, so, so this is one way to invoke comparison. This is sometimes a little hard to make work because sometimes it's hard to find the right series that does the job, I mean, this is sort of a natural candidate to compare to, but, and then we have another version of the comparison test, so the limit comparison, I'm going to put this up and then I'll put it back down, so the limit comparison test, again, I need positive terms, so both things are positive, I guess I'll still write it again, A, N, P, N are both positive things, I guess, well, okay, so if the limit as N goes to infinity of A N over B N is not zero and is defined, so it's not zero, well, let's say N exists, well, we say it's some number, so there's some number that this limit approaches, then A N, B N converge and they diverge together, they do the same thing. Now, it's important to think about what this is saying because if you just kind of memorize the test, you'll get it wrong, so limit comparison sort of saves you a little bit because the things look kind of sort of the same, but one's not bigger, one's not smaller, maybe they sort of wiggle around one above the other, right? A to U have, here's your comparison series and here's the thing you want to compare to. Well, this one is never bigger than, I mean, sometimes it's bigger, sometimes it's smaller, so I can't use direct comparison, but the limit is the same. So the limit comparison is just saying it doesn't matter what happens at any finite time, just what happens in the end. And this statement that the ratio goes to some number means eventually, this one is L times that one, so you have to remember and interpret what this is and not just try and apply it just mechanically because there are other tests that we will do like what I'm going to introduce today where there's a limit going on and if it's zero, it's fine but if it's one, it's very bad. So here you have to think about what this limit is telling you rather than just say, there's something with a limit, I do the limit and then I don't, that doesn't give you time. Okay, so that's where we stand and so I just reviewed what we did so far, let me continue a little bit. So one thing to notice is that other than a few special series, if some of the terms are negative, we don't know what to do. All right, so the integral test is only good for positive terms, the p-series is only good for positive terms because p-series all have positive terms. Comparison is only good for positive terms. So sometimes series have negative terms in them and we do anything with that. Well, let's look at the simplest case, for example the geometric series, it's okay if it's negative, right? So for a geometric series like minus one-third to the n, this is a geometric series because there's a ratio there of one-third, we know that this is one over one minus minus one-third, which is four-thirds, three-quarters. But this certainly has negative terms because it's one minus one-third plus a ninth minus a twenty-seventh plus an eighty-first minus blah-blah-blah. So this term, this one, the techniques that we developed other than we already know about geometric series don't apply. The integral test doesn't tell us anything, the comparison test doesn't tell us anything, but there's positive and negative terms here. Well, in fact, some of these guys with positive and negative terms are actually easier. So here's another example, say, which is not a geometric series, minus one over n to the n-th power is one to infinity. So this is, did I get the sign on here? No, okay. So this is one minus one plus minus a third plus a quarter minus, oops, yeah, minus a fifth, et cetera. But this converts or diverges, you don't know. So we can't apply any of these tests that we have on the board here other than the first one, which is just mess around and see what happens. How many people think this converges? It's okay, you can have an opinion. I see four, maybe 10. How many people think this diverges? I see a couple. How many people, their arms don't actually work? One. Okay, so how many people, like, went by Starbucks or Dunkin' Donuts before they got here and got some coffee? Okay, you need to go get coffee and then come back, okay? You're like, how? Okay, so what can we do about something like this? I don't know, you're the teacher, you tell me. Well, we don't have any tools to do it, but let's just think about what's going on by doing physics instead. Let's just think about what's going on by looking at what's happening with the suns. So, we start with negative one, I'm going to draw a picture over here of what I get. So, this is the first sum, shoot, all right, it's going to be negative, so I'll put zero up here. So, I start here with the first term, my sum is negative one. My next term is minus one plus a half, which is here and a half. So, it goes up. My next term is minus one plus a half minus a quarter. So, it drops down halfway to where it was before. Well, my scale's a little off to minus three quarters. Then, S4 is minus one plus a half, oops, except I can't do arithmetic, this is a third. So, third is more than a quarter, so it's a little lower, but it's still higher than this. And I add back a quarter. Well, a quarter is less than what I took away last time. So, I move up, but I don't move up higher than I was before. And then, I think you see the pattern. When I take away a fifth, I move down, but I don't even move down lower than what I was before. And so, I have this up, down, up, down, up, down motion, or it's like a pendulum that's swinging back and forth and eventually it seems to wind down. It's going to do this kind of thing. So, I mean, I can prove this explicitly, let me not, because it will take the rest of the class, or at least another 20 minutes, and you'll all, you need more space than you are now. But, how many people think now that we can do this? More hands work. Wow, it's amazing. Okay. So, this converges to something. So, what is the pattern that makes this go? Suppose I do something instead of that. Suppose I try something similar. Well, okay, so what is it that makes this go? Yeah? It's alternating. Okay. Okay, so it's alternating. Okay, what about this one? Suppose I try that one. That was alternating too. This is minus 1, plus 2, minus 3, plus 4, minus, oops. How about let's just do this. Plus 4, minus 5, plus 6. Is that what we're going to converge? No, yes. Right. Right. So, it's alternating, and the amount of the alternation is less each time. It's just like a pendulum. You have a pendulum that's swinging back and forth, and eventually it winds down. You could also have a pendulum that's swinging back and forth, and it gets bigger and bigger and bigger, and then it isn't going to converge. So, we have two things here. It's alternating, and the alternations become less and less. So, this leads us to yet another test. Right now, we just call it the alternating series test. Which is really easy. So, if I have a sum, so I don't like that either. How about we say, yeah, okay, that's fine. So, always every step I change time. They go from plus to minus, plus, minus, plus, minus. As n goes to infinity of an is 0, then the alternating series, what we just emphasized, it's an alternating series, converges. Now, a series of stuff where the absolute values go to 0, which is the same 0, and it alternates sums, plus, minus, plus, minus, plus, minus, then it goes, then it converges. And in fact, the amount that changed, you can also, just from this behavior, which is more general, you can also figure out how close you are. To the limit. Because, okay, I'm not, if I stop here at S4, I'm not off by more than a fifth. Because a fifth is the biggest deviation I'm going to make, because it's an alternating series. The terms are decreasing. So, the fifth, so I guess I mean it's decreasing, too. So, I'm going to throw this part in here, then it's decreasing. So, to be a convergent alternating series, we want to say that it's decreasing. It only really needs to be eventually decreasing. The reason that we want it to be decreasing is so that we can also say how much we're off by if we add up 100 terms. So, if I do, so, so here's a little more that comes from it. If I want to know, so that the difference between the limit, the sum up to infinity, and the sum stopping at some term N is no more than the absolute value of the term you didn't use. Just say this one. So, this guy is called the alternating harmonic series. So, the one that diverges, the one that holds positive terms is called the harmonic series. This one is called the alternating harmonic series. So, how many terms, let's do the alternating harmonic series, minus 1 to the N over N, starting at 1, I need to get the answer within, so if you understood everything I've said so far, you should just be able to tell me the answer. I heard somebody say it, yeah, yeah, 99, right. Because the mistake, if I add up, well sometimes it's an N plus 1. If I just start adding these things, then this is the sum within the next, the error is at most 1 over 100, all right? It's off by no more than 1 over 100. I don't know, it's 1 over 99, I guess it's odd. So, yes, as I move it closer. So, I switch, I'm starting at negative 1, I'm just starting at plus 1, so it's okay. If I could just go to up and down and up and down, and the oscillation that they add is tiny. And it doesn't go away from the actual sum of my orders and this, so that's easy. Now, you know, in the homework you could do a few alternating series, but this is easy. So, there's one, so we can, but not all series are alternating series, so suppose I have something, maybe I'll use a little bit, suppose I have something like, something like the sum of the cosine of N over N, make an N squared, N is in radians. So, would I expect this to converge? Why? Cosines are bigger than 1, so it's less than 1 over N squared, but it's not all positive terms, right? I mean, I don't know actually what the cosine of 1 is or the cosine of 2 or the cosine of 3 off the top of my head, but this is not all positive. So, according to the tests that I have so far, I don't know anything. It's not an alternating series because the cosine doesn't go plus, minus, plus, minus. It goes plus, plus, minus, minus, minus, plus, minus, plus, plus, plus, minus, plus, plus, plus, minus. It bounces all around, so it's not an alternating series. It's like an alternating series, but it doesn't quite alternate. So, I can't really use any of these things that are on the board here, except the, just mess around and see what happens. But this definitely converges because if I were to take absolute values by comparison, and that's what you want to do. So, if I take absolute values and I get something that converges, then it doesn't change what happens. If I change some of the signs to minus, because it only makes it better. So, in fact, so this sort of gives us another test. So, do I have enough room? Sure. You'll raise how many terms it is going to be. Really a definition, but if I take absolute values and it converges, so does the guy without absolute values. We think about it. If I take absolute values and it converges, that's changing all the terms to positive. Or if I change all the terms to negative, then those both converge. So, if the positive works, then all negative will work, too. And then if I just change some of the terms, it's only going to make it better. Because if I change some of the terms to negative, if I take all the terms to positive, and change a few to negative, well, it will make it smaller. Now, in fact, this is so useful and so common that we give this a name. So, we say that this series converges absolutely, or is absolutely converging. So, we have an infinite sum, and if the sum of absolute values converges, then the series is called absolutely converging. This utility of this name absolutely converging, as we will see soon, is sometimes you want to know if there's a variable in there, and you don't care if that variable is positive or negative. It's absolutely converging, then both happens. We'll see soon. Okay? So, this idea of absolutely converging sounds hard, but it's easy to just make all the terms positive and make the sum. So, there are some series which are absolutely convergent, and some series which are not absolutely convergent but convergent. So, as an example, this guy, this series, alternating 1 over n squared is absolutely convergent. Because if I forget about this alternating piece and just make it a 1, I still get a convergent series because it becomes a p series with p equals 2. But there are some that are not absolutely convergent, but this one is not absolutely convergent, but it converges, because if I take absolute values, I get the harmonic series which diverges. But this is an alternating series, and it's limited to 0, so it converges. So, this situation is called conditionally converging. So, we have three non-situations. We have absolutely convergent, conditionally convergent, and divergent. If something is absolutely convergent, then both the series and its absolute value converge. If it's conditionally convergent, the series converges, but its absolute value doesn't. And if it's divergent, well, then it's divergent. Okay? We have these three situations. So, the alternating series test tells us about conditional convergence. Do I need to do any more examples of alternating series? I think it's very straightforward. You look at it, if the signs go bit-bop, bit-bop. Okay. So, I want to get to one more test of convergence which is extremely important, because you use it all the time. We'll use it like all the time in the next bit of the class. And the idea is, let's look again at geometric series for a minute. So, here we have a geometric series. Now, what can we take as a specific number? Not sure. I'll just write them down. Sixteenths, sixty-fourths, blah, blah, blah. And, in fact, let's multiply it by three. Make it, I'll make it, I'll multiply it by 11 so I don't have to change any number. There. So, this is a geometric series. And it converges to, it's 11 times 1 over 4 to the n, n equals 0 to infinity. So, this converges to 11 over 1 minus a quarter. So, it converges to 44 over 3. Okay. That's fine. But, what's going on with this geometric series? So, most of these tests that we've done you have to find a friend to compare it to other than the integral test which just looks at the series. But, with the geometric series, notice that each term is some fraction of the previous term. So, if I take the next term and I divide it by the term before in a geometric series, then this ratio is a number. If I divide 11 over 64 by 11 over 16, I get a quarter. If I divide 11 over 16 by 11 over 4, I get a quarter. If I divide 11 over 4 by 11, I get a quarter. If I look at the ratio of one term to the next term, it always goes down by a specific amount. So, this gives us another way to look at a series. And see whether it's shrinking fast enough to have a whole converging. Right? So, if we look at this ratio, and it's a small number, then we would expect the series to converge because it's shrinking faster than a geometric series. And, let's leave the absolute value for fun. So, we know this and this will converge. This will converge if the absolute value of r is less than 1. So, if the ratio of the geometric series is small, then the series converges. But, maybe we can apply this to something that isn't the geometric series. Let's take a series like that the other stuff doesn't really. So, let's take an easy example. Suppose that I want to sum up 1 over n factorial. So, remember n factorial, so n factorial means n times n minus 1 times n minus 2 on down to 2 times 1. That is, take n and all the stuff less than it and multiply it all together. Well, so far, none of these tests on the board apply to that. I don't know how to integrate 1 over n factorial. I guess I can do a comparison to it, actually, because eventually I could do some kind of comparison and make it work. But, let's just look at it directly. So, this is 1 plus a half plus 3 times 2, 4 times 3 times 2, so this gets small pretty fast. And, this state that I made, if I compare this to a geometric series, this shrinks even faster than a geometric series because I'm multiplying it by something smaller than a geometric series. Every time the terms go down, not by a number r, but they go down by a number 1 over n. So, they go down faster than a ratio. And, if we look at the limit, and we really need the limit. So, if I look at the next term divided by this term, well, this is 1 over n factorial, oops, over 1 over n factorial. So, if I look at the nth term, n plus first term and divided by the nth term, we get that, which if I do a little algebra. So, this is n factorial over n plus 1 factorial. And, since n plus 1 factorial is just n plus 1 times, so n factorial is n times n minus 1, n minus 2, that's a bit of a bit. n plus 1 factorial is n plus 1 times n times n minus 1, that's a bit of a bit of a bit. So, this is really n factorial divided by n plus 1 times n factorial because the factorial is just multiplied by the next higher, that's all the lower number. So, this is just 1 over n plus 1. So, as I go further and further into the series, I'm multiplying by a more vanishingly small number. So, this shrinks faster than a geometric series, than a geometric series. So, if I were to do a comparison to a geometric series, I should see that it would converge. So, if I take the limit as n goes to infinity of this, then I get the limit as n goes to infinity, 1 over n plus 1, which is 0. So, this should converge because ultimately, it's like a geometric series with a ratio of 0. So, let me write that up to something where maybe the ratio is 1.5 or something, but let's just write that. Let me write that as a test, if you will. So, this goes by the name of the ratio test because you look at the ratio of one term to the next. The ratio test says that if the limit as n goes to infinity of the next term over the current term is some number r, well, let's put absolute values. If the absolute value of the ratio is some number r, then this, then we have three possibilities. If r is less than 1, it converges for sure. The series converges absolutely. Because we're doing absolute values, we get absolute convergence. If the ratio is bigger than 1, well, then this is bigger than a geometric series with a ratio bigger than 1 which diverges. So, that means that this series diverges. And then there's a third possibility. If the ratio is 1, you don't know. So, if the ratio is 1, this test is no good to you. Try another thing. So, one reason, so the ratio test is probably the most important, the most useful test for convergence. We use it a lot because what we're going to do, well, you'll see. So, we're going to use the ratio test a lot. And it's not too hard to use provided you can do a limit. One thing I want to emphasize here, here there's a limit involved. And here, if the limit is 0, that's good. And if the limit is 1, that's bad. If you're doing limit comparison, and the limit is 0, that's bad. And when the limit is 0, you compare to a bad thing. So, it's important to keep these straight in your mind what the limit is telling you. So, a lot of times, people don't even teach limit comparison because things get confused. If you get confused, then throw away limit comparison. But better if you don't get confused.