“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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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

泰勒级数

在最后一个模块中，我们介绍泰勒级数。与从幂级数开始并找到其代表的函数的更好描述不同，我们将从函数开始，并尝试为其寻找幂级数。无法保证一定会成功！但令人难以置信的是，许多我们最喜欢的函数都具有幂级数表达式。有时，梦想会成真。和许多梦想相似，多数不说为妙。我希望对泰勒级数的这一简介能激起你学习更多微积分的欲望。

- Jim Fowler, PhDProfessor

Mathematics

Centered around a.

[SOUND] Thus far, we've been talking about McLoren series or

about Taylor Series centered around 0.

But I can write down a power series for a function not centered around 0,

but centered around some other point a.

What I mean is, I could try to write the function f

as a power series, n goes from 0 to infinity, some coefficient C sub n,

not just times x to the n, but times (x-a) to the nth power.

Well, just like before, let's assume that f has a power series representation and

then try to figure out what those coefficients must be.

Let me be a little bit more precise, right?

What I'm saying here is, I'm going to assume this holds whenever x is, say,

within big R of a.

And assuming this is true, I want to figure out what these coefficients have to

be in terms of the function f.

So let me just write down the first few terms, right?

When I plug in n = 0, I just get C sub 0 times 1,

because it's something to the 0th power, + when I plug in n = 1,

I get C sub 1 (x-a) to the 1st power + when I plug in n = 2,

it's C sub 2 times (x- a) squared, and then it keeps on going.

Now, what happens when I plug in a for x?

So that's asking what is f(a).

Well, in that case, it's C sub 0 +, what's this term?

It's just 0, +, what's this term?

Well, it's just 0 +, all the other terms are 0.

So this is telling me that C sub 0 is equal to f(a).

Well, another way to say that is that if I knew that this was true for

values of x within big R of a, then I know what C sub 0 has to be.

C sub 0 has to be the value of f at the point a.

The next coefficient can be computed in terms of the derivative.

But not the derivative at 0, but the derivative at the point a.

Well, let's differentiate this and see what we get.

So I'm assuming that this is true whenever x is within big R of a.

So the same will be true for the derivative at least.

So the derivative of f(x) will be the sum, n goes not from 0 but

from 1, to infinity of the derivative of this,

which is C sub n times n times (x-a) to the n-1.

And this is true at least when x is within that same big R of a.

Okay, let me write down the first few terms here.

Let's plug in n = 1, I get just C sub 1 times 1 times (x-a) to the 0th power,

so that's just C sub 1.

+, when I plug n = 2, I get C sub 2 times 2 times (x-a) to the 1st power.

When I plug in n = 3, I get C sub 3 times 3 times (x-a) squared,

and then it keeps on going.

And now note what happens when I plug in a for x.

All right, what's f'(a)?

Well, it's just like what happened before, I've got C sub 1,

but then this C sub 2 times 2 times (x-a), that term vanishes.

This term here also has an (x-a), so when x is equal to a,

this term vanishes, all the later terms vanish.

This means that I can recover the value of the C sub 1 coefficient,

just by differentiating the function at the point a.

And so on, right, just like before, I can now write down a formula for

the nth coefficient.

So in general, what I'll find is that C sub n can be computed by taking

the nth derivative of f at the point a and dividing that by n factorial.

So, let me summarize all of this, the Taylor series for a function f centered at

a, or sometimes you'll see just a Taylor Series around a, for the function f.

It's given by this,

it's the sum n goes from 0 to infinity of the nth derivative of f at a.

That's the big difference, divided by n factorial.

Here's another difference, (x-a) to the nth power.

And it sort of makes sense, right, the Taylor Series around a is

a power series around a, so it's got this (x-a) to the nth term.

But instead of calculating the derivative at 0,

we're calculating the derivative at the point a.

[SOUND]

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