“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

Radius, zero.

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Nobody's guaranteeing us all that much convergence.

As an example let's consider this power series.

The sum n goes from 0 to infinity, of n factorial times x to the n.

For sure, if x is equal to 0, then this power series converges.

Yeah so what happens when x is equal to 0?

Then this series is the sum, n goes from 0 to infinity,

of n factorial times 0, to the nth power.

[LAUGH] Now what is this sum?

All right. Does this series converge?

Well, I prefer the convention that 0 to the 0 is 1.

And I think pretty much everyone prefers the convention that 0 factorial is 1.

So, the first term, the n equals 0 term of this series is equal to 1.

What about the next term, what about the n equals 1 term?

Well, that's 1 factorial times 0 to the first power, that's 0.

What about the next term, what about n equals 2?

That's 2 factorial times 0 squared, that's 0.

What about n equals 3, that's a number times 0 to the 3rd power, that's 0.

What about n equals 4, that's a number times 0 to the 4th power,

that's equal, all the other terms in this series are 0.

So, this series has the value 1.

It converges at x equals 0.

But are there any other values of x, besides 0,

for which this series converges?

So, I know this converges at x equals 0,

but let's suppose that x is some non-0 number.

Then does this series converge or diverge?

Well let's check it with the ratio test.

So I'm going to look at the limit as n goes to infinity.

What's the n plus first term here?

That's n plus 1 factorial times x to the n plus 1 divided by just the nth term.

Which is n factorial times x to the n, I'm looking at the absolute value of that.

What's this limit?

Well I can simplify this limit.

Right this is the limit of, what's n plus 1 factorial over n factorial?

Well, that I can simplify a bit.

Right that's just n plus 1, and then I've got x to the n plus 1 over x to the n.

That's just x.

So for some fixed value of x, which isn't 0, what is this limit?

Well if x is anything but 0, this limit is enormous number times non-zero number.

This is very, very positive, right?

This limit is infinite and that is bigger than 1.

What that means is that by the ratio test this series diverges.

For any fixed value of x not 0, this series diverges.

So it doesn't converge anywhere else.

It only converges when x equals 0.

So, in other words, since this series only converges at the point x equals 0 and

diverges whenever x is not 0, that means the radius of convergence is equal to 0.

Let me leave you with a question.

Try to think of other power series with the radius of convergence equal to 0.

Can you think of any other examples?

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