“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

Holograms.

[SOUND] They tell me that if you take a little bit of a hologram,

and you look through it, you can still see the whole picture.

The same is sort of true of analytic functions, all right?

Let's think about sine.

Here, I've got the graph of a function, y = f(x), and

in this case, the function that I'm graphing is just sine.

I want to think about this graph a little bit, and

just focus on this region right around the origin.

Imagine, if you will, that I've taken away all of the graph,

and I'm just looking at this little tiny piece right around the origin.

What can I figure out about sine just by looking right around the origin?

Well, if I look around the origin,

I can figure out the value of sine at the origin.

If I'm just looking around the origin, I can figure out the derivative of sine

at the origin, I can figure out the second derivative of sine at the origin.

I can figure out the third derivative of sine just by looking at this little tiny

piece of the graph.

I can figure out all of the derivatives of sine just by looking at this

little tiny piece of the graph of sine.

And if I can calculate all the derivatives of sine, that means that just by

looking at that piece, I can write down the Taylor series for sine at 0.

And we've already seen that the Taylor series for

sine converges to sine everywhere.

That's exactly right.

We've seen that sine of x is equal to its Taylor series for all values of x.

And what that means is that just this little tiny piece of sine just around

the origin manages to encode the entire story about sine.

All of the values of sine can be recovered just

by knowing a little tiny piece of sine around the origin because that's enough to

calculate all the derivatives at 0, which is enough to recover the Taylor series,

which recovers the entire function.

Contrast that with something involving absolute value.

Well, here's the graph of a function, so y = f(x).

But in this case, the function is f(x) = |x-1|-1.

Now what happens if I zoom in on the origin?

Well this little piece of the graph just around the origin is enough

to calculate some facts about f, right?

What can I calculate just by looking right around the origin?

Well, I can calculate that the value of this function is 0, when I plug in 0.

I can calculate that the derivative of this function is -1 at 0,

and I can calculate that all of the higher derivatives, all of the second, third,

fourth, and so on derivatives at 0 are equal to 0.

So, yeah, this thing looks like a straight line around here.

And that infinitesimal information at the point 0, just the derivatives at the point

0, are enough to recover the function around 0, our whole interval around 0.

Now of course, I might have been tricked into thinking

that the entire function just looked like this, right?

I can't tell that there's this point where the function fails to be differentiable

just when I'm looking over here.

But nevertheless,

just this infinitesimal information at a point is still enough to recover some

information about the function in a whole interval around that point.

Real analytic functions are really quite surprising.

Infinitesimal information just at a point is giving you information

on a whole interval, sine, cosine, and e to the x, or even better.

In that case, the Taylor Series converges to those functions everywhere, which means

infinitesimal information at a single point on, say, the graph of sine, right?

A single point on the graph of cosine or at a single point on the graph of e to

the x, that information is telling you everything about the entire function.

Yet, and entire is actually the word that's used.

Entire describes functions like sine, where just the infinitesimal

information at a point is enough to recover a power series with an infinite

radius of convergence, converges everywhere, to the entire function.

So it's really like a hologram, right?

Just a little tiny bit of a hologram records everything in the scene.

And so too would, say, sine of x.

Just a little, tiny piece of sine reveals everything about the function.

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