“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

Analyticity.

[SOUND] Suppose f is a function and

suppose that I can compute the derivative of f at 0,

the second derivative of f at 0, the third derivative of f at 0,

the fourth derivative of f at 0, and so on.

Suppose that I can compute the nth derivative of f(0),

regardless of what n is.

So in short, suppose that f is infinitely differentiable at 0,

meaning that, although it might be really hard, at least in principle,

I could compute the millionth derivative at 0, the billionth derivative at 0.

I could compute any higher derivative that I want, at the point 0.

Well, then the power series for f around 0 as the sum n goes from 0 to infinity,

of the nth derivative of f at 0.

This is the thing that makes sense, because f is infinitely differentiable.

Divided by n!, times x to the n.

But what I don't know is whether that power series converges to f

when x is near, but not equal to 0.

What I'm asking is whether this Taylor series is actually

equal to the function, right?

What's the relationship between this Taylor series and

the original function that I'm studying?

Sometimes, it happens that the function is equal to its Taylor series.

Let's give a name to that phenomena.

Here's how we're going to talk about this.

The function f, which I'm assuming to be infinitely differentiable,

is said to be real analytic at the point 0 if there's

some number R positive so that this happens.

So that the function is equal to its Taylor series,

maybe not everywhere, but at least when x is within R of 0.

You can make this a bit more general.

Well, let's think about what this is really talking about, right?

What this definition is trying to get at is the idea that the function is equal to

its Taylor series around 0.

That's what real analytic at 0 means.

It means the function has a power series representation around 0.

Well, to generalize this, I can talk about power series around other points.

So here's the definition of what it means for a function to be real analytic

not just at zero, but real analytic at some arbitrary point a.

Well, it should still mean that there's some big R,

which will measure how far x is from a now, instead of from 0.

And here I've written down the Taylor series around 0, but

I should write down the Taylor series around a.

So instead of differentiating at 0,

I'll differentiate f n times at evaluate it at a, and

then I'll multiply not just by x, but by (x- a) to the nth power.

And then I don't want to say that x is close to 0,

I want to say that x is within R of a.

So this is saying that, at least near a,

when you're within R of a, f can be written as a power series.

And that's what it means to say that f is real analytic at a point a.

Let me draw a diagram, to try to convey what is going on here.

Let's suppose that this piece of paper represents all of the functions,

from the real numbers to the real numbers.

And plenty of those functions are just miserable functions.

Here's a graph of a terrible looking function,

it's got a lot of discontinuities.

Some of these functions, though, are continuous, so

within the collection of all functions, I've got the continuous functions.

Here's a graph of what looks to be a continuous function.

Still not great because it's still got these spikes but at least it's continuous.

Now some of these continuous functions are differentiable.

A function that's differentiable is necessarily continuous.

So within the collection of all continuous functions,

there's smaller collection of just the differentiable functions.

They don't have these terrible spikes.

But just because you're differentiable doesn't mean that you're,

let's say, twice differentiable, or three times differentiable.

So within the collection of all the differentiable functions there's

an even smaller collection of functions, which are infinitely differentiable.

These are functions that I can differentiate once, twice, three times.

These are functions I can differentiate as many times as I like.

Sometimes people call these functions smooth functions or c infinity functions.

This is a very restrictive class of functions.

But there's an even more restrictive class of functions.

Within the smooth functions, there's a smaller collection of functions,

the real analytic functions.

These functions aren't just smooth,

right, it's not just that I can differentiate these functions.

These functions also have the property that if I write down their Taylor series

around some point, that Taylor series converges near that point to the function.

So these real analytic functions are really quite special.

I mean, not every function is real analytic and yet the surprise is that so

many of the functions that we care the most about turn out to be real analytic.

[SOUND]

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