“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

数列

欢迎参加本课程！我是 Jim Fowler，非常高兴大家来参加我的课程。在这第一个模块中，我们将介绍第一个学习课题：数列。简单来说，数列是一串无穷尽的数字；由于数列是“永无止尽”的，因此仅列出几个项是远远不够的，我们通常给出一个规则或一个递归公式。关于数列，有许多有趣的问题。一个问题是我们的数列是否会特别接近某个数；这是数列极限背后的概念。

- Jim Fowler, PhDProfessor

Mathematics

Let's list all of the real numbers.

[MUSIC]

Let's make this even easier on ourselves.

Instead of trying to list off all the real numbers,

let's just list off the real numbers between zero and one.

And what do I mean by that?

I mean, I want a sequence so that every single real number between zero and

one appears somewhere in that sequence.

Well, suppose that I did.

Suppose that this is the list.

This is the first number in my list, the second number in my list, and so on.

And just imagining that I have a list of all the real numbers between zero and one.

Using this sequence, I'll build a number x.

So here's how I'll define this number x.

I'm going to define this number x by telling you what the nth digit

after the decimal point of this number's supposed to be.

And I'm going to do that in terms of this sequence, a sub n.

So let's suppose D is the nth digit of a sub n.

Then the nth digit of x is either one more or one less than D.

How to be one more than D if D is less than or equal to seven.

And how to be one less than D if D is eight or nine.

We'll see how this goes.

So here is the sequence that I'm supposing as a sequence of

all the real numbers and let's start by trying to write down this number x.

So x is zero point and

I don't know what the first digit is after the decimal point and

to do that I'll look at the first digit of the first number and that's an eight, and

if it's an eight, then my digit is going to be one less, which is seven.

To get the next digit, I'll look at the second digit of the second number,

and that's a two.

And two is less than or equal to seven, so I'll make that a three.

To get the next digit, I'm going to look at the third number in my list,

and the third digit is the third number seven.

And that means third digit of x is going to be an eight, and

I can keep on playing this game, and if I keep on doing this

I’ll eventually write down a number that stars off like this, 738 and so on.

Supposedly I listed off every single number between zero and one on that list.

X is also a number between zero and

one, so it must be one of the numbers on our list.

Well, here's our list and here's x.

Is x the first number on our list?

No, it's not our first number because the first digit after the decimal point of

the first number is an eight.

But the first digit after the decimal point of x is a seven.

All right, I'm using this formula for determining the digits of x.

So the first digit of the first number is an 8,

and that means the digit for x is a seven.

Well, maybe x is the second number on our list.

It can't be that one,

because the second digit of the second number on our list is a two.

But using this formula for the digits of x, the second digit of x is a three.

And because x differs from a sub two in the second place after the decimal point,

x can't be a sub two.

Maybe x is a sub three.

Well, the third digit after the decimal point for a sub three is a seven, and

the third digit after the decimal point for x is a n eight.

X and a sub three differ in the third place after the decimal point.

Maybe x is a sub four.

X and a sub four differ in the fourth place after the decimal point.

The fourth digit here is a four and the fourth digit of x is a five.

Maybe x is a sub five.

X and a sub five differ in the fifth digit after the decimal point.

The fifth digit after decimal point is a seven.

But the fifth digit of x is an eight.

So x can't be a sub five.

Ca it appear anywhere on our list?

No!

But why not?

Well, the number x isn't a sub n.

It isn't the nth number on our list,

because x differs from a sub n in the nth digit.

Remember how we defined the number x?

D was the nth digit of a subscript n, and x was defined in

such a way that the nth digit of x was definitely not D.

And because the nth digit of x isn't the same as the the a sub n,

X can't be a sub n.

X can't be any of the a sub n's.

So, x cannot appear on the list.

So what does this mean?

It means there's no sequence which mentions all of the real numbers between

zero and one.

If there were such a sequence, I could use that sequence to build the number x,

which would have to be on the list,

because it's a list of all real numbers between zero and one, and yet

can't be on the list because it differs from the nth number and the nth digit.

That means that it's impossible to list off all the numbers between zero and one.

That means there's no sequence which mentions

every number between zero and one.

In a sense then, there's more real numbers than there are integers.

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