“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

Infinite radius.

[MUSIC]

It can certainly happen at the radius of convergence is infinite.

Well let's consider this power series.

The sum n grows from 0 to infinity of x to the n divided By n factorial.

Let's try the ratio test.

So here we go, the limit n goes to infinity of the n plus first term.

So x to the n+1 over (n+1)!

divided by this the nth term.

So let's display here Xn over n factorial, and absolute

value of that because I'm checking for absolute convergence with the ratio test.

I can simplify this a bit.

This is a fraction with fractions in the numerator and denominator.

This is the limit n goes to infinity of Xn + 1 times n

factorial in the denominator of the denominator, put that in the numerator.

Divided by x to the n times n plus 1 factorial.

So, I've just got a fraction, but I can simplify this a bit, too.

This is the limit n goes to infinity.

We've got x to the n plus 1 over x to the n,

just leaves you with x in the numerator.

And I've got n factorial in the numerator and n plus 1 factorial in the denominator.

Well, n plus 1 factorial kills everything here, right?

N plus 1 factorial is 1 times 2, all the way through n plus 1.

And that contains all the terms and n factorials.

So what I'm left with is just an n plus 1 in the denominator.

Now, what is this limit?

X is just some fixed quantity.

It doesn't depend on n.

But what's the limit then of sum number x divided by n + 1, n going to infinity.

Well this limit is 0.

It doesn't matter what x is.

If you take some fixed number divided by a very large quantity,

you could make this equals to 0 as you like.

So this limit is 0, 0 is less than 1.

And that means by the ratio test this series converges regardless of what x is.

So what's the radius of convergence?

Series converges for all values of x.

And that means the radius of convergence is infinity.

And in the not-too-distant future, we're going to see a very surprising result.

We're eventually going to see that this series

is in fact a complicated way of writing down a function we already know.

This is just a complicated way of writing down the function e to the x.

Meaning that if you plug in a specific values for x say, x equals negative 1.

You get that the sum n goes 0 to infinity of -1 to the n

over n factorial is equal e to the -1 is equal to 1 over e.

And by choosing value of x, by plugging in different specific values of x.

We can generate a ton of really neat series.

Here's another example.

Just the fact that the sum N goes from zero to infinity of ten to the N

over in factorial is well, according to this, that's just E to the 10th power.

And that is really cool.

But let me warn you and share a bit of the philosophy of power series with you.

Yes. By plugging in specific values of X,

you can generate a ton of interesting examples but

power series aren't just a way of generating a bunch of series in isolation.

Part of the joy of power series comes by thinking of power series not

as a mechanism for generating a bunch of discreet examples.

But as a way of collecting together a whole bunch of interesting series that

depend on a parameter x.

[SOUND]

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