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

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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?

[MUSIC]

As we've seen, practically all of the

convergence tests that we have at our disposal

have made the assumption that the terms in

the series are positive or at least non-negative.

I want to prove that the sum n goes from 1 to

infinity of minus 1 to the nth power over n squared, converges.

So, how am I going to do this?

I can't apply the usual convergence tests because not all the terms

are non-negative.

I mean look, here I've written out some of the terms, minus

1 plus a fourth, minus a ninth plus a 16th minus a 25th.

I can only apply the comparison test if the

terms were non-negative, and that's not the case here.

But I can think about absolute convergence.

I can use the theorem that

absolute convergence implies just regular old convergence.

So if I can just prove that this series converges

absolutely, then I know that it converges in just the usual sense.

So let's try that.

What I know is this.

The sum n goes from 1 to infinity of the

absolute value of minus 1 to the n over n squared.

Well, that's exactly the same thing as the sum n

goes from 1 to infinity of just 1 over n squared.

And that's a p series, with p

equals 2, and because 2 is bigger than 1, this p series converges.

Now, what does that mean then about the original series I care about?

That means that the sum n goes from 1 to infinity of minus 1 to the n

over n squared, converges absolutely because some of the

half loop values converges and consequently, by the theorem,

it just plain old, converges. That's often how this is going to work.

Let's suppose you want to analyze this series, the sum

n goes from 1 to infinity of a sub n.

You've been given this task.

Well, the first thing I'd suggest you do is the limit test.

Take a look at the limit of a sub n as n

approaches infinity, because if that's not 0, then you know your series diverges.

You're done.

But let's suppose the series passes that test,

then what do you do?

Well, then you can hope that the terms in the series,

the a sub n's are all greater than or equal to 0.

because if you've got a series, all of whose terms

are non-negative, then you can apply all the usual convergence tests.

But if that's not the case, if you're in a situation where some of

these terms are positive, some of these terms are negative, what do you do?

Well, then I'd recommend that you apply all

of our old convergence tests, not to this

series directly, but to this series.

The sum n goes from 1 to infinity of the absolute value of the a sub n's.

What I'm suggesting that you do is

try to prove that this series converges absolutely.

Because if you know that this series converges absolutely,

then you know this series just plain old converges.

So, absolute convergence is an important idea

not because every single conversion series converges absolutely.

That's not even true.

There are series that converge but don't converge absolutely.

Nevertheless, a lot of series do converge absolutely.

So an easy way to prove convergence, is to prove absolute

convergence and then to use the theorem that absolute

convergence just implies regular old convergence.

That's going to be successful a lot of the time.

[NOISE]

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