Calculus Two: Sequences and Series is an introduction to sequences, infinite series, convergence tests, and Taylor series. The course emphasizes not just getting answers, but asking the question "why is this true?"

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From the course by The Ohio State University

Calculus Two: Sequences and Series

899 ratings

The Ohio State University

899 ratings

Calculus Two: Sequences and Series is an introduction to sequences, infinite series, convergence tests, and Taylor series. The course emphasizes not just getting answers, but asking the question "why is this true?"

From the lesson

Alternating Series

In this fourth module, we consider absolute and conditional convergence, alternating series and the alternating series test, as well as the limit comparison test. In short, this module considers convergence for series with some negative and some positive terms. Up until now, we had been considering series with nonnegative terms; it is much easier to determine convergence when the terms are nonnegative so in this module, when we consider series with both negative and positive terms, there will definitely be some new complications. In a certain sense, this module is the end of "Does it converge?" In the final two modules, we consider power series and Taylor series. Those last two topics will move us away from questions of mere convergence, so if you have been eager for new material, stay tuned!

- Jim Fowler, PhDProfessor

Mathematics

Coarseness. [MUSIC] Often we've been asking for things on the nose. Like in this formulation of the comparison test. For all n, 0 less than equal to a sub n less that equal to b sub n, if this series the sum of the b sub n converges, then this series the sum of the a sub n converges.

When you formulate it this way, I'm asking for this inequality to hold on the nose that, a sub n be between 0 and b sub n for all n. But now, we're beginning to see, that we don't need to be quite so rigid. Well, one thing we've seen is that convergence depends only on the tail of the series. So, something like this, this assumption here for all n, 0 is less than or equal to a sub n is less than or equal to b sub n. This can be weakened. I can get away with just assuming this, that for all large values of little n, a sub n is between 0 and b sub n. And I can replace the statement that, a sub n is less than b sub n, with something with the limit of the ratios. I'll still make the assumption that the terms are non-negative, but instead of this inequality, I could rewrite that to just say that, a sub n over b sub n, this ratio is between zero and one.

And the point I'm thinking about this ratio, or I could just get rid of this and instead of thinking about this ratio being between zero and one, I could just ask that the limit of this ratio be some finite value L. And at that point, [LAUGH] I've written down one direction of the limit comparison test. So, we've replaced the comparison test, which is really a pretty rigid statement, with something much more flexible. So, yeah, the limit comparison test is a coarser version of just the regular old comparison test. [LAUGH] And by coarser, I actually am claiming, this is a sort of philosophical claim, that the limit comparison test is actually better than the Comparison Test. And it's better for two reasons. Well, first of all, the limit comparison test is better than just the comparison test in practice, all right? Beyond any sort of theoretical advantage, I mean the limit comparison test just ends up being easier to apply. It's easier to verify this condition, than it is to cook up some sequence of b sub n's which satisfy this and the sum of the b sub n's converges, say, right? This is much harder to satisfy than just some limit statement like this.

Well, let's see and expose an example in practice where, where this really is much easier. So, here's an example of a series. Let's analyze the convergence of this. The sum n goes from 1 to infinity, of this fraction n squared plus the square root of n minus n divided by n to the 4th plus n squared minus n. Does this series converge or diverge?

What I can do is build a sequence out of this. So, I'm going to call the terms here, a sub n, and then I'm going to look at this other sequence, sequence b sub n equals 1 over n squared. And I'm going to compute, the limit of the ratio between the terms in these sequences. So, what's the limit as n goes to infinity, of a sub n over b sub n? Well, dividing by 1 over n, is the same as just multiplying by the reciprocal. So, I'm really asking about this limit. But this isn't so painful now, right? because now, I'm multiplying n squared times this. So, this limit is this fractions limit, this fractions, n to the 4th plus lower order terms n squared times n squared plus lower order things, and the denominator is the same, it's just n to the 4th plus lower order of things, in this case n squared minus n. So, what's this limit? Well I have got, n to the 4th plus lower order terms divided by n to the 4th plus lower order terms, this limit ends up being 1. So, the limit comparison test applies and [LAUGH] what I know, is that this series, the sum of the b sub n's converges because this is just the p series with p equals 2. And that means as a result of the limit comparison test, I can then conclude that this very complicated series also converges. But it's not just that it's easier to apply, right? There's a second reason, why this kind of coarse thinking is really beneficial? And the reason is that thinking this way actually gets us closer to understanding what convergence really means. Alright? What's conversions all about? Well, it's certainly not about the first million series, right? Convergence only depends on the tail. And what convergence really depends on is, how quickly the terms are decaying to zero. So, something like the limit comparison test, that asks you to look at the limit of the ratio of the terms in the series, is really asking you to think about how quickly those terms are decaying to zero. And that's good, right? That means the limit comparison test isn't just a way to get the answers more easily. The limit comparison test is, is really showing us something about what convergence really means.

And that's the goal of mathematics. The goal of mathematics isn't just answers. It's understanding. Well, think back to that example again, alright? This example wasn't just answering a question. This example is demonstrating this philosophy, right? That, I'm doing more than just answering the question of this series convergence. I'm saying that there's a reason, an understandable reason why this series should converge. This series converges because it resembles, in the sense of this limit between the terms in these series, but this series resembles 1 over n squared. And this p series, where p equals 2 converges, it's where the limit comparison test this series does too. But it's not just a calculation. That's not just an answer. That's a reason, why something that looks like this ought to converge [SOUND].

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