“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

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

It's time for the harmonic series.

[MUSIC]

That series has a name.

This infinite series, the sum of the reciprocals of the positive whole numbers,

is called the harmonic series.

I want to know if that series diverges or converges, and

the first thing to do is to look at the limit of the terms.

The nth term of the harmonic series is 1/n, so

if I take the limit as n goes to infinity of the nth term,

well, whats the limit of 1/n as n goes to infinity?

This is 0.

And what did the limit test say?

The limit test tells us that if the limit of the nth term of some series

is not equal to 0, then the series diverges.

But when the limit is equal to 0, [SOUND].

So in this case, the limit of the terms is 0, and the limit test is silent.

At this point, we still don't know whether the harmonic series converges or diverges.

I can start adding up a bunch of terms.

1 + 1/2, the first two terms of the harmonic series, = 3/2.

If I add the next term to the harmonic series, 1 + 1/2 + 1/3, that's 11/6.

And I could add the next term on the harmonic series, 1 + 1/2 + 1/3 + 1/4.

Well, that's 25/12.

I could add the next term in the harmonic series, 1 + 1/2 + 1/3 + 1/4 + 1/5.

That's 137/60, and I could keep going like this.

If I add the first ten terms together, 1, 1/2, 1/3, 1/4,

and so on up to 1/10, I get a number that's just under three.

Let's add up even more terms.

So instead of just ten terms, I'll add up the first 100 terms and

I'll get a number, 5.187 or so.

The question is what happens when I add up more and more terms in this series?

Even more terms.

Well, if I add up the first 1000 terms, I get a number that's about 7.485.

I add up the first 10,000 terms, I get a number that's just under 10.

We're adding up a ton of terms and still, the partial sums just aren't that big.

Let's take a different approach.

Here, I've started writing out the harmonics series, 1 + 1/2 + 1/3.

I've written down the first 32 terms of the harmonics series and, of course,

it keeps going.

I'm going to group the terms in a clever way.

I'll set aside the 1 and this first 1/2.

And then I'll put these two next terms together, the 1/3 and the 1/4.

I'll put these next four terms together, 1/5 + 1/6 + 1/7 + 1/8.

Then I'll put these next eight terms together, the terms between 1/9 and 1/16.

And I'll put these next sixteen terms together, between 1/17 and 1/32,

and so on.

Now let's underestimate each of these groups.

I can underestimate 1/3 by replacing 1/3 with something smaller than 1/3, like 1/4.

1/4 is smaller than 1/3.

I can underestimate 1/5, 1/6, and

1/7 by replacing each of those with something smaller than 1/5, 1/6, and 1/7.

Well, an 1/8 is smaller than 1/5, an 1/8 is smaller than 1/6, and

an 1/8 is smaller than 1/7.

Why is this a good idea?

Well, here I've got two 1/4s, and two 1/4s is 1/2.

And that means 1/3 plus 1/4, if I add those together, is at least 1/2.

Here, I've got 4/8.

4/8 is also 1/2.

And that means 1/5 + 1/6 + 1/7 + 1/8, which is bigger than 1/8 + 1/8 + 1/8.

1/5 + 1/6 + 1/7 + 1/8 must be bigger than 4/8.

It must be bigger than 1/2.

I've got eight more terms here, starting at 1/9 and ending at 1/16.

Each of these terms can be underestimated by a 1/16.

1/9 is bigger than 1/16.

1/10 is bigger than 1/16.

1/11 is bigger than 1/16.

1/12 is bigger than 1/16.

All of these terms are bigger than 1/16.

But now I've got 8/16.

And that means 1/19 + 1/10 + 1/16, all of these together, must be at least 8/16.

And 8/16 is 1/2.

So if I add up these eight terms in the harmonic series I get at least a half.

Now the next 16 terms in the harmonic series are each at least 1/32.

But if I add up 16 terms, each of which is at least 1/32,

that's an answer which is at least 1/2.

The next 32 terms in the harmonic series, starting at 1/33 and

ending at 1/64, those 32 terms are each at least 1/64.

But 32/64 is 1/2, and

that means the sum of the next 32 terms in the harmonic series is at least 1/2.

The next 64 terms in the harmonic series, starting at 1/65 and ending at 1/128.

Those 64 terms added up is at least 64/128.

It's at least 1/2.

So when I'm adding up the harmonic series, I can underestimate the answer

by adding 1 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2, and so on.

So summing the harmonic series is even worse than adding up 1/2 and

1/2 and 1/2 and 1/2.

All that is to say is the harmonic series diverges.

Let's summarize this.

The harmonic series diverges even though the size of the terms gets very small.

Even though the limit of the nth term is 0.

And this isn't a contradiction.

The fact that we know is that if a series converges,

then the limit of the nth term is 0.

But just because the limit of the nth term is 0, doesn't mean the series converges.

The harmonic series is a great example of this phenomena,

where the limit of the nth term is 0, but the series, nevertheless, diverges.

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