“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 put all the integers together.

[MUSIC]

The goal here is to come up with a sequence that mentions every single

integer.

But before we do that,

let's first try to come up with a sequence that mentions every non-negative integer.

It's the sequence a sub n = n where the index n starts at 0.

So the terms of the sequence are 0, 1, 2, 3, and so on.

And of course, I could negate that sequence to get a sequence that mentions

every negative integer.

So I can look at the sequence, b sub n = -n, but let's start n at 1.

And if I do that, then the terms of this sequence

start- 1,- 2,- 3,- 4 and so on.

But I want both.

I want a single sequence that includes among its terms every positive integer,

every negative integer, and zero.

Is it even possible?

Yes, I'll weave together the two sequences that we've already built.

So what do I mean?

Well let's take a look at these two sequences.

I could put them together, I could weave them together.

I could start with 0, then do- 1, then 1, then- 2, then 2, then- 3.

All right, I get a new sequence that would end up mentioning every single integer.

It would start 0,- 1, 1,- 2, 2,- 3,

3,- 4, 4, and it would keep on going like that.

I'd like a formula for that sequence.

Well, here is a formula for the sequence.

C sub n will be defined via this piecewise definition depending

on the parity of n, whether n is odd or even.

If n is odd, then the nth term will be negative n + 1 over 2.

And if n is even, then the nth term is just n over 2.

And I'll start with the index 0.

And this will give me this sequence.

This 0th term, when I plug in 0, 0 as even, 0 over 2 is 0 and

that gives me this 0.

When I plug in 1, 1 is odd, so I get -1 + 1 over 2, that's -1.

When I plug in 2, that's even, so it's 2 over 2, that's this 1 here.

When I plug in 3, 3 is odd, so it's- 3 + 1 over 2, that's -2.

And it just keeps on going like that.

There's another way to think about this.

So say that I've got the same quantity of dots and squares,

is really to say that there’s some method by which I can pare off

the dots and squares so that every square gets a dot and every dot gets a square.

And once I pair them off like this it’s very believable

that there’s the same quantity of dots and squares.

Well something similar is going on with non-negative integers and all integers.

But let us take a lot at the non-negative integers.

I perhaps want to show that there's the same quantity

of non-negative integers as there are just all integers.

And to do that, I just have to tell you some method for

pairing off non-negative integers with all of the integers.

And that's exactly what this sequence does, it assigns to 0 the number 0.

It assigns to 1 the number- 1.

It assigns to the index 2 the number 1.

To the index 3 the sequence assigns the number- 2.

To the index 4 it assigns the number 2.

To the index 5 it assigns the number 3.

To the index 6 it assigns the number 3, and it will keep on going.

And eventually, we've ended up pairing off

every single non negative integer with every single integer.

That really means that there's the same quantity

of non negative as there are all integers.

That should really give you pause, that sounds impossible.

Think about some physical object, some finite object like coffee beans.

If I've got some coffee beans but

then I take some away, now I've got fewer coffee beans.

But the collection of all integers is different.

If I start with all the integers, and just take away from the negative integers,

I got the same quantity of things.

Why does that work?

Well one definition of an infinite quantity,

is a quantity that needn't get smaller, even when you take something away.

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