“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

Finding the radius.

[SOUND] Let's suppose that I've

been given a power series.

Perhaps it's the power series the sum n goes

from 1 to infinity of x to the n divided by n squared.

How do I find the radius of convergence.

Well let's try the ratio test.

So I'm going to look for absolute convergence.

So I'm really trying to figure out for which values of x that series converges.

And I'm going to look at the ratio of the m plus the first term over the nth term so

the m plus first term x to the n + 1 over (n + 1) squared.

And I'm going to divide that by the nth term,

which is exactly what I've got there, x to the n over n squared.

Now, I can simplify that fraction a bit.

So this is the limit n goes to infinity.

And I've got x to the n plus 1 over x to the n.

So I'll just write absolute value of x.

And then I've got (n +1) squared but it's in the denominator of the numerator.

And I've got n squared in the denominator of the denominator.

So I can write this as n squared over (n+1) squared.

Now, what is this limit?

Well when n is very large, this quantity here is very close to one and

this x doesn't depend on n at all so this limit is just the absolute value of x.

And this is the ratio between the n plus first and the nth term, so

to get absolute convergence of this series,

it's enough for the ratio test that this be less than one.

But I also know something about when the series diverges.

By the ratio test, when this limit, which is the absolute value of x,

when that limit is bigger than 1, then this series diverges.

So putting it all together, what's the radius of convergence?

So to think about that, let's draw a diagram.

Here we've got a number line.

And what I know is that when the absolute value of x is less than one,

then the series converges absolutely.

So that tells me that the series converges when x is between -1 and 1.

I also know that when the absolute value of x is bigger than 1,

then the series diverges.

That tells me the series diverges when x is bigger than 1 and

the series diverges when x is less than -1.

So it converges in between here diverges to the right of this and

diverges to the left of this.

Now admittedly I haven't thought about what happens at x equals

-1 when x equals 1 but I don't need to if

all I care about is knowing the radius of convergence, all right.

I'm thinking about this interval being where the power series converges,

and maybe it converges at minus 1, maybe it converges at 1.

But what's the radius of this interval?

Now this is an interval centered at 0 and it's radius is 1.

And that tells me that the radius of convergence is 1.

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