“微积分二：数列与级数”将介绍数列、无穷级数、收敛判别法和泰勒级数。本课程不仅仅满足于得到答案，而且要做到知其然，并知其所以然。

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From the course by The Ohio State University

微积分二: 数列与级数 (中文版)

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“微积分二：数列与级数”将介绍数列、无穷级数、收敛判别法和泰勒级数。本课程不仅仅满足于得到答案，而且要做到知其然，并知其所以然。

From the lesson

交错级数

在第四个模块中，我们讲解绝对和条件收敛、交错级数和交错级数审敛法，以及极限比较审敛法。简而言之，此模块分析含有一些负项和一些正项的级数的收敛性。截至目前为止，我们已经分析了含有非负项的级数；如果项非负，确定敛散性会更为简单，因此在本模块中，分析同时含有负项和正项的级数，肯定会带来一些新的难题。从某种意义上，此模块是“它是否收敛？”的终结。在最后两个模块中，我们将讲解幂级数和泰勒级数。这最后两个课题将让我们离开仅仅是敛散性的问题，因此如果你渴望新知识，请继续学习！

- Jim Fowler, PhDProfessor

Mathematics

Conditional convergence.

[MUSIC]

Not every convergent 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 a 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 (-1) to the nth power / n.

I know that series is not absolutely convergent.

Well I know this series is not absolutely convergent for

the following reason, right?

I can look at the sum n goes from 1 to

infinity of the absolute value of (-1) to the n / n.

Well, what is that?

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

What's the absolute value of (-1) to the n?

That's just 1, this is just 1 / 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, it does converge.

And in fact, the sum n goes from 1 to infinity of (-1) to the n / n,

ends up converging to- 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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