“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

The tale is what matters.

[MUSIC]

If you go out and read the literature on conversions tests and the like,

you'll sometimes find that those theorems are written in a way

that's a bit more vague than we're used to.

So the comparison test, if a sub n is between 0 and b sub n for

all n, and the series sum of the b sub ns converges,

then this series, the sum of the a sub ns converges as well.

Do you see what's missing?

Well, I didn't write the sum n goes from 1 to infinity of b sub n, or

the sum n goes from one to infinity of a sub n.

But this is actually okay.

Here's a theorem.

Let big M be a natural number.

Then this series, the sum little M goes from big M to infinity of a sub m

converges if and only if this series,

the sum of little m goes from 1 to infinity of a sub m converges.

In short, convergence depends on the tail.

So I've got this series, the sum little m goes from 1 to infinity of a sub m, and

I can start writing it out like this.

It's a sub 1 plus a sub 2, blah blah blah,

plus a sub big m- 1 plus a sub big m plus a sub big m + 1 + and so on.

And I can take out my scissors of mathematics and

cut this series like this.

This piece here I'll call the head, and what's left over, which is really

the sum little m goes from big M to infinity of a sub m, this is the tail.

And what the theorem tells me is that this series converges if and

only if this series converges.

So, convergence only depends on the tail.

Why does convergence only depend on the tail?

Let's suppose that this tail converges.

That means something about the sequence of partial sums.

What that means is that if I consider the sequence of partial sums

s sub n equals the sum little m from big M to n of a sub little m.

That sequence of partial sums converges to something,

I'll call it big L, that's the value of this series.

I can relate those partial sums to the partial sums for

the series that begins with a sub 1.

So now let's think about this series, right,

the series m goes from 1 to infinity of a sub m.

The value of this series is by definition the limit of the partial sums for

that series.

And that's the limit little n goes to infinity of the sum little

m goes from 1 to little n of a sub m.

Now I can analyze this in terms of the series that starts at big M.

All right, what is this limit?

Well that limit is a sub 1 plus dot, dot,

dot plus a sub big M- 1 plus this limit.

The limit little m goes to infinity of the sum little m from big M to limit

n of a sub m.

All right, both of these are just the limits of the sum

of the terms a sub 1 through a sub n.

But now I'm assuming that this limit exists,

because I'm assuming that that series that starts at big M converges,

that's the limit of the partial sums for that series which I was calling L before.

Which is just to say that this series has a finite value, right.

This limit exists.

And what that means is that if I suppose that the tail of

a series converges, then the original series converges as well.

Here's the upshot.

If you only care about convergence, writing down this, just the sub over n of

a sub n, is just as good as writing down this more complicated looking thing,

the sum little n goes from 1 to infinity of a sub n.

because all I care is about convergence.

I don't actually need to know where the series is starting.

Because regardless of where I start the series, they all converge or

they all diverge.

It only depends on the tail.

[SOUND]

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