“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

Welcome to week five of sequences and series.

[MUSIC]

We're going to be looking at power series.

These are series that look like this.

The sum, say N goes from zero to infinity of A sub N,

just some numbers, times X to the N.

That means you get to pick a sequence, A sub N, for example,

maybe a sub N is two to the N, just the sequence of the powers of two.

The important thing here is that A sub N, doesn't depend on X in any way.

A sub N is just a formula on this case given in terms of N but not X.

And from that sequence, you build the power series.

Or continuing with this example, if A sub N is two to the N,

then the associated power series is the sum and

goes from zero to infinity of two to the N times X to the N.

The cool thing is that in a ton of cases the power series that we're building

are actually representing functions we already know about.

Or what about this case?

What happens here?

[LAUGH] Well, one over one minus X is the sum and

goes from zero to infinity of X to the N.

That's just the formula for summing a geometric series.

Now look what happens if I replace X by two to the N.

Then one over one minus two X.

That must be the sum, N goes from zero to infinity of two X to the N,

which is exactly what I've got here.

This is the sum N goes from zero to infinity of two to the N times X to the N.

So, this mysterious seeming power series is actually just a complicated way of

writing down this very reasonable seeming function, one over one minus two X.

The other cool thing is that power series are like polynomials.

If I just write down the first few terms, right,

I could just look at the first few terms of this series, 1 + 2X

+ 4X squared + 8X cubed, if I truncate this series I just get a polynomial.

And then as son as I write the plus dot, dot, dot,

as soon as I'm thinking of this as sort of a polynomial that goes on forever,

well that's really what a power series is.

So power series are really very cool.

They let us translate a lot of our intuition for

polynomials into other more complicated functions.

We're going to see that there are power series representations for

very complicated functions, like sign and cosign.

But since these power series look like polynomials, it's going to let us

translate some of that intuition about polynomials into those more complicated,

transcendental functions.

[SOUND]

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