“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

级数

在这第二个模块中，我们将介绍第二个主要学习课题：级数。直观地说，将数列的项按照它们的顺序依次加起来就会得到“级数”。一个主要示例是“几何级数”，如二分之一、四分之一、八分之一、十六分之一，以此类推的和。在本课程的剩余部分我们将重点学习级数，因此如果你在有些地方感到疑惑，将会有大量时间来弄清楚。另外我还要提醒你，这个课题可能会令人感到相当抽象。如果你曾经为此困惑，我保证下一个模块提供的实例会让你感到豁然开朗。

- Jim Fowler, PhDProfessor

Mathematics

What is a series?

[MUSIC]

We're trying to capture with some precise mathematics and intuitive idea.

I've got a list of numbers imagining that I want to add up all of these numbers.

But the list goes on forever.

Well I could just start adding.

Here's the first number plus the second number plus the third number in my list.

Just start adding up all of the numbers in my list.

But if I'm really going to add up all of the numbers, I'd never finish.

So what am I supposed to do?

We'll use partial sums.

What's a partial sum?

Instead of trying to consider this series where I add up

all of the a sub k's, right.

With this infinity up here.

And instead I'm going to consider the partial sum.

I'm just going to add up, the first n terms of this sequence.

And I'll call that s sub n.

Lets see this a bit more concretely.

If i wanted to calculate say, S sub five the fifth partial sum.

Add up the first five terms of my series.

Add up A sub one, A sub two, A sub three, A sub four and A sub five.

If I were to calculate S sub seven, the seventh partial sum,

well I'd be doing the same thing but I'd be adding up the first

seven terms of my series.

This is potentially a very confusing point.

I've got a sub k's, and s sub n's.

I mean, I started with the sequence a sub k.

And out of this sequence, I built the series.

And from this series, I then started considering another sequence.

The sequence of partial sums built out of this series,

which was itself constructed from this sequence of numbers.

So why is this sequence of partial sums useful?

Well here's why.

I want to add up all of the terms in the series.

But I'll never finish that task.

So instead the partial sums are telling me just to add up a lot of the terms.

S2 is just the sum of the first two terms, which I could compute.

Then I could compute the sum of the first three terms, s3.

Then I could compute the sum of the first four terms, I could compute the sum of

the first five terms, I could compute the sum of the first six terms.

I could compute the sum of the first hundred terms,

the sum of the first thousand terms, the sum of the first million terms.

And if I add more and more terms, hopefully I'm getting closer and

closer to what would happen, if I added up all of the terms in the series.

To trick then is to take a limit.

I'm never going to finish adding up all of the terms, but I can add up lots and

lots of terms and see if I'm getting close to anything in particular.

Here's the official definition.

If I want to add up all of the numbers in my list, all of the a sub k's and

I'm going to take a limit of the partial sums.

I want to hit that limit of adding up the first n terms in my sequence.

There's a bit of terminology to introduce.

If the limit of the sequence of partial sums exists and

is equal to finite number L, then we say, that the infinite series converges.

What if limit doesn't exist or what if the limit is infinity?

If the limit of the partial sums doesn't exist or

it's infinity, then I say, that the series diverges.

All of this is setting up the basic question that'll occupy us for

the rest of this course.

Given a series.

Does it diverge or does it converge?

And if converges, what is it converge to?

[SOUND]

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