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

898 ratings

The Ohio State University

898 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

Absolute convergence. [SOUND]. What do I even mean by absolute convergence? Well here's the definition. Right, so defintion,

the series, I'll write down a series, the sum n goes from 1 to infinitiy of a sub n Converges absolutely. So this is the term that I'm defining. Defining absolute conversion. So I'm going to say the series converges absolutely if what happens?

If the series, just the sum n goes from 1 to infinity of the absolute values of the a sub n's just plain old converges in the usual sense. What's an example of this? While ago, so that this series. The sum n goes from 1 to infinity of the absolute value of sine n over 2 to the n; that series converges. you could prove that by doing a comparison test with the geometric series 1 over 2 to the n. Alright, so that series converges, and that means that this series, the sum n goes from 1 to infinity of just sine n over 2 to the n, no more absolute value now, just sine n over 2 to the n. That series converges absolutely, or we might say is absolutely convergent, just because this series, the sum of the absolute value of this quantity, converges. But with a name like absolute convergence, you would hope. That an absolutely convergent series is also just convergent in the regular sense of converging. So that's a theorem. If the sum of the absolute values of the a sub n's converges, then just the sum of the a sub n's converges. In other words, if a series converges absolutely then just plain old converges, let's prove that. Okay yes so how's this proof going to go. Well the proof is going to start out with my assuming that the series converges absolutely. Meaning I'm going to assume that the sum of the absolute value of the a sub n's converges. And then somehow you know, I want to do something. I don't know what yet, so I've got some cloud with question marks. And then, I eventually want to conclude that the original series converges. So what goes in these question marks, right?How do I get from this assumption to this conclusion. Well the first thing I can note is that if this series converges Then the sum of twice those terms also converges.

Why is that important? Well here is the key fact, this is sort of a trick if you like. But what it's saying is that 0 is less than or equal to a sub n plus the absolute value of a sub n is less than or equal to twice the absolute value of a sub n.

Why is this even true? Well, think about the possibilities. Maybe a sub n is negative. And if a sub n is negative, then I've got a negative number plus well, the absolute value of that negative number. These two things cancel, and I just get zero. And then certainly this is true. Zero is less or equal to zero, is less than or equal to twice some positive number.

Well what if a sub n were positive? Well, then I'd have 0 is less than or equal to a positive number plus itself, cause the absolute value of a positive number is just that same number, is less than equal to twice that same number. Well, in that case, these are equal right, I've got a positive number plus itself. That's the same as twice that positive number. So if a sub n's positive, this is true if a sub n's negative, this is true if a sub n's 0, this is just 0 less than or equal to 0 plus 0 less than equal to 2 times 0. So whether a sub n is positive, whether it's 0, whether it's negative, no matter what happens, this is a true statement.

Now we can apply the comparison test. So by the comparison test this series converges right. The sum of these converges the side angles from 1 to infinity of a sub n plus the absolute value of a sub n. Well that's term wise less than this convergent series and all the terms are non negative so that means this series converges. but now our original series can be written in terms of this series and the sum of the absolute values in the a sub n's. So what I got is this series converges, this series converges, but the difference of these two series is the series that I'm originally interested in, right. And the difference of convergent series converges so that mean that the series just the sum of the a sub n's converges. Which is exactly what I wanted to show. So we prove the theorem. Alright, if a series converges absolutely, then it just plain old converges. And that really justifies the terminology. I'm calling this, well everybody calls this, absolute convergence, because it's a strong kind of convergence. Absolute convergence implies convergence, so absolute convergence is even stronger than just regular old convergence. [SOUND]

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