“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

Let's think about a more complicated series.

[MUSIC]

How can we approach this series?

It's the sum k goes from 0 to infinity of sin of k squared, divided by 2 to the k.

Does this series diverge or converge?

Looks suspiciously similar to a series that we do understand.

Yeah, it looks a bit like this series.

All right, if I just got rid of this sin of k squared term and

replaced it with a 1, I'd have this geometric series.

This series converges and it's value is 2.

Let's be careful.

I want a precise relationship between the mysterious series that I don't understand

and the geometric series that I do understand.

Precisely right, how are these related?

Well, sin of k is between minus 1 and

1 so, sin of k squared is between 0 and 1.

And now if I were to divide all of this by 2 to the k, look,

now I'd have sin of k squared

divided by 2 to the k is between 0 and 1 over 2 to the k.

What's the original goal?

What am I actually trying to understand here?

I'm trying to understand whether this series converges or diverges.

But what does that question mean?

That question really means that I'm supposed to

be looking at the sequence of partial sums.

I should add up the terms between k equals 0 and n, and, then,

ask about the limit of the partial sums.

And if this limit exists,

well that's exactly what it means to say that this original series converges.

What sort of sequence is s sub n?

Well one thing I know about the sequence of partial sums is that the sequence of

partial sums is nondecreasing.

How do I know that?

Well the terms that I'm adding up look like this.

And those terms are non negative.

So, if you add a non negative number, the result's not getting any lower, right?

And that means that the sequence of partial sums is not getting any smaller,

it's nondecreasing.

Not only that, the sequence of partial sums, s sub n is also bounded.

So why bounded?

Well it again boils down to this inequality.

Because this is less than this,

if I add up a bunch of these that's less than adding up a bunch of these.

It's exactly what I'm saying here.

If I sum all of these terms it's less than or equal to the sum of these terms.

Because each of these terms is less than or equal to each of these terms.

And that's just because sin squared of k is less than or equal to 1.

But I know something else here, right?

This is a convergent geometric series.

As I let n drift off to infinity, this is approaching 2.

So putting this all together, I've got that these partial sums are bounded by 2.

So I've got a bounded monotone sequence.

But a bounded monotone sequence converges.

So we know something about the original series.

And because the sequence of partial sums converges,

that's exactly what it means to say that the series converges.

This idea, this idea of taking a series we do understand and

using that series to explore a series we don't understand, this idea has a name.

It's called the comparison test.

This idea is important enough that in a future video,

we're going to write down a specific statement of the comparison test.

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