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

Conditional convergence.

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Not every convergence series converges absolutely.

Seeing that absolute convergence and just plain old convergence

are related, right, absolute convergence implies plain old convergence.

But it turns out that this doesn't go the other way.

It's not the case that convergence implies absolute convergence.

So we'll give a name to this situation.

So here's the definition, a series is conditionally convergent

if the series converges, but the sum of the absolute values diverges.

In other words, conditional convergence means

the series converges but not absolutely.

Let me draw a diagram of the situation.

So if I start out considering all series, once I start thinking

about convergence, right, that separates series

into two different kinds of series, right?

The divergent series and

the convergent series.

But now I've got this more refined notion of convergence, absolute convergence.

So I can subdivide the convergent series into two kinds of series,

those that are absolutely convergent, and

those that are just conditionally convergent.

I mean, they still converge, but they don't converge absolutely.

Now, can I think of anything in

the conditionally convergent part of that diagram?

Can I think of any conditionally

convergent series at all? Well, here's an example.

The sum n goes from 1 to infinity of negative 1 to the nth power divided by n.

I know that series is not absolutely convergent.

Well I know this series is not absolutely convergent, for the following reason.

Alright?

I can look at the sum, n goes from 1 to infinity

of the absolute value of negative 1 to the n over n.

Well, what is that?

That's just the sum n goes from 1 to infinity.

What's the absolute value of minus 1 of, to the n?

That's just 1, this is just 1 over n, this

is just the harmonic series, and the harmonic series diverges.

And since the sum of the absolute values diverges, it's exactly

what it means to say that the series does not converge absolutely.

But the series does converge.

Yeah, does converge and

in fact the sum n goes from 1 to infinity of minus 1 to the

n over n ends up converging to negative the natural log of 2.

And since this series does converge, but it doesn't converge absolutely.

This is an example of a conditionally convergent series.

We don't quite yet have the tools to show that, but we're awfully close.

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