“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

Welcome to week six of sequences and series.

[MUSIC]

Well thus far, we've been starting with a power series and then asking the question,

what function does that power series represent?

For example, we considered this series,

the sum n goes from 0 to infinity of x to the n.

And we ended up showing that this series is equal to 1 over 1 minus x,

provided that the absolute value of x is less than 1.

We've also built a lot of tools for

transforming one power series into a new power series.

For instance, I can differentiate a power series term by term.

So in this case, what do I get?

Well, if I differentiate this term by term,

I get the sum n goes from 1 to infinity of n times x to the n-1.

That's the derivative of x to the n with respect to x.

And that's the derivative of 1 over 1-x.

And we calculate the derivative of 1 over 1-x, well that's 1 over 1-x squared.

So there we've got it.

I've got a power series and I found a function that represents that power

series, at least on this interval when the absolute value of x is less than 1.

So that's what we've been doing.

We've been starting with a power series, and

we've been trying to identify what function does that power series represent.

And then we would be transforming those power series,

messing around with them by differentiating them term by term,

integrating them term by term, multiplying them together, things like that.

But this week we're going to turn all of that around.

What I mean is that so far in the course, we've been starting with a power series.

And then from that power series, we've been getting an iso description of

the power series as some function that we're familiar with.

And I want to turn that around now.

I want to start with a description of a nice function, and then try to find

a power series representing that function on some interval.

[SOUND]

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