“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。 注意：此课程的注册将在2018年3月30日结束。如果您在该日期之前注册，您将可以在2018年9月之前访问该课程。

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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。 注意：此课程的注册将在2018年3月30日结束。如果您在该日期之前注册，您将可以在2018年9月之前访问该课程。

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

There is more terminology.

[SOUND] What is a geometric progression?

A geometric progression is a sequence with a common ration between the terms.

We should see an example. Maybe the sequence starts 3, then 6,

12, 24, 48, 96, and it keeps on going.

And let's say the general rule for

the sequence is an = 3 times 2 to the n.

Why is that a geometric progression?

Well, there's a common ration of 2 between each of these terms.

To get from 3 to 6, I have to multiply by 2.

To get from 6 to 12, I multiply by 2.

To get from 12 to 24, I multiply by 2.

To get from 24 to 48, I multiply by 2.

All right, that's the common ratio between all the terms in the sequence, it's 2.

We can write down a general formula for a geometric progression.

So I can write an = the first term,

a0, times the common ratio r to the nth power.

In this particular example, a0, the first term, is 3, and the common ratio is 2.

Well, here's a question.

Why are these things even called geometric progressions?

Well in a geometric progression, each term is the geometric mean of its neighbors.

Okay, but what is a geometric mean?

Well, the geometric mean of two numbers,

of a and b, is defined to be the square root of ab.

Why is a geometric mean called geometric at all?

What's geometric about it?

Well, here's one geometric story you could tell yourself.

Could build a rectangle, one of whose sides is a and

the other side has length b, then this rectangle has area ab.

And I want to build a square whose area is also ab, what's its side length?

Well, the side length will be the square root of ab.

So this is some kind of geometric sense in which an average of a and

b might deserve to be the square root of ab, a geometric average.

So the deal with geometric progressions is that each term is the geometric mean

of its neighbors.

So let's see that in our original example, 3, 6, 12, 24, and so on.

The claim is that in a geometric progression,

each term is the geometric mean of its neighbors.

Let's see that here.

What's the geometric mean of 3 and 12?

Well, it's the square root of 3 times 12.

That's the square root of 36, that's 6.

So yes, 6 is the geometric mean of its neighbors.

Let's try it again with 12.

What's the geometric mean of 6 and 24?

Well, that's the square root of 6 times 24, 6 times 24 is 144.

And the square root of 144 is 12, so yeah, 12 is the geometric mean of 6 and 24.

The limit of a geometric progression depends very strongly on that

common ratio.

In our example here, what's the limit as n approaches infinity of an?

It's infinity, because I can make an as big as I like,

provided that I choose n big enough.

What if the common ratio were 1/3?

Well, here's an example of a geometric progression with common ratio of 1/3,

1, 1/3, 1/9, 1/27, 1/81, and so on.

What's the limit of an in this case, as n approaches infinity?

Well, that's really the limit as n approaches infinity of 1/3

to the nth power, because that's a formula for the nth term in this sequence.

Well, that limit is 0, right?

By making n big enough, I can make an as close to 0 as I like.

Other interesting things can happen too.

You should think about what happens when that common ratio is negative.

[SOUND]

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