“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

幂级数

在第五个模块中，我们学习幂级数。截至目前为止，我们一次讲解了一种级数；对于幂级数，我们将讲解整个系列取决于参数 x 的级数。它们类似于多项式，因此易于处理。而且，我们关注的许多函数，如 e^x，也可表示为幂级数，因此幂级数将轻松的多项式环境带入棘手的函数域，如 e^x。

- Jim Fowler, PhDProfessor

Mathematics

Convergence depends on x.

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Let's consider a power series.

So maybe I've got a power series, the sum n goes from 0 to infinity

of some coefficients, a sub n times x to the nth power.

Now suppose that I know that, that power series converges absolutely, when x = 3.

So yeah, I'm going to assume that it converges absolutely at x = 3, and

then I want to know, what about our other points?

What about at, say, x = 2?

Does it converge there?

Well then, it must converge when x = 2.

Let's see why.

Well since it converges absolutely at x = 3, that just means that the sum,

n goes from 0 to infinity of the absolute value of a sub n times 3 to the n, right?

This series converges.

We can compare this to the same series when x = 2.

But what I mean the say is just that, 0 is less than or

equal to the absolute value of a sub n times 2 to the n, which is less than or

equal to the absolute value of a sub n times 3 to the n.

And since this series, the sum of a sub n times 3 to the n converges,

that means by the comparison test this series,

the sum n goes from 0 to infinity of a sub n times 2 to the n, this series converges.

Which is just to say that the original series,

when x = 2, well in that case,

this series converges absolutely.

And of course, there's nothing special about the number 2.

So if x is any value, so the absolute value of x is less than or equal to 3.

That just means that x is in [-3, 3].

If x is any value in that interval, then 0 is less than or

equal to the absolute value of a sub n times x to the n.

Well that's just because it's the absolute value of something.

But then, that is less than or

equal to the absolute value of a sub n times 3 to the n.

So again, by comparison, all right,

that means that this series, the sum n goes from 0 to infinity of,

I'll just write, a sub n x to the n converges absolutely.

Because the sum of the absolute values converges,

because I'm comparing with this convergent series.

And this is the usual case.

This is usually what happens.

To talk about this, though, let me be a little bit more formal.

Let me give a name to this.

Let's call C the collection of all real numbers,

so that this power series converges.

Or in words, C is all the real numbers x, so that this series converges.

It's a collection of numbers.

The big deal here is that C is an interval.

Well here's the theorem.

This collection of values of x where the power series converges,

it turns out that collection of points is an interval, by which I mean,

maybe it's this open interval, maybe it's a closed interval.

Or maybe it's something more complicated, like some half open interval.

We'll see a proof of that soon.

And since it's an interval, this collection of points,

where the power series converges, is called the interval of convergence.

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