“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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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

Alternating series.

[SOUND] What is an alternating series?

For example, this series,

the sum n equals 1 to infinity of -1 to the n plus

1 over n is an example of an alternating series.

What does that even mean?

Let me write down some of the terms I think will be a little bit clearer.

So if I plug in n = 1, I get -1 squared which is 1 over 1.

So 1 over 1.

When I plug in n equals 2, I get -1 cubed over 2, that's -1 over 2,

so that's minus a half.

When I plug in n equals 3, that's negative 1 to the fourth, over 3,

that's 1 over 3, that's plus a third.

When I plug in n equals 4, I get negative 1 to the fifth,

which is negative 1 over 4, so minus a fourth.

When I plug in n equals five, I get negative one to the sixth,

over five, that's 1 over 5, so plus a fifth.

When I plug in n equals six, I get negative one to the seventh, over six, so

that's just negative 1 Over 6, so minus a 6th.

And then it's going to keep on going like that.

But I'm flip-flopping between these two colors.

I'm alternating signs, right?

I'm adding, subtracting, adding, subtracting, adding, subtracting.

It's an alternating series.

So that's what I mean by alternating.

Well, here's a precise definition.

The series, sum a sub n,

n goes from 1 to infinity,

is called an alternating series.

If a sub n = (-1) to the nth power * b sub n.

And all the b sub n are the same sign.

So maybe the b sub n sequence is they're all positive.

And then a sub n has this term here, times a positive sequence.

Well, this term is what makes the signs, the S-I-G-Ns, flip-flop back and forth.

Well here's a bit of a warning.

Not every series which has both positive and

negative terms is an alternating series.

For example, this series, the sum n goes from 1 to infinity

of -1 to the n times sine n over n.

This is not an alternating series.

I mean, yes, it's got the -1 to the n here.

But the sine of n also introduces its own quite complicated pattern of positive and

negative terms.

So this is a series, some of the terms are positive, some of the terms are negative,

but it's not alternating, right?

This is not an alternating series.

A couple of other examples.

Sum n goes from 1 to infinity, say -1 to the n times n plus 1 over 2.

So that's all in the exponent there, n times n plus 1 over 2, say all over n.

This is also not an alternating series.

And it's still got a minus 1 there to raise to some power but

the power's a little bit more complicated.

N times n plus 1 over 2, at terms or some of them maybe positive, some of them maybe

negative but they're not flip flopping in sign, in S-I-G-N back and forth.

In contrast here's an example that is an altering series,

the sum n goes from one to infinity say of -1 to the n times sine squared n over n.

Sine squared n, this turns out to always be positive, right?

The n is positive here and -1 to the n.

This is the only thing that's affecting the S-I-G-N of this term.

This does indeed in flip flop back and forth between negative positive,

negative positive, negative positive.

This is an alternating series.

[SOUND]

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