“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

Not all series converge.

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Let's suppose that a series converges.

So let's suppose that the sum, k goes from one to infinity

of a sub k is equal to L.

Informally what that means is that the limit

as n approaches infinity of the finite some K goes

from one to n of ak = L.

Right, to say that infinite series equals L is to say the limit of the partial

sums is L.

I can modify this slightly, without effecting the limit l.

What I mean, is that the limit as n approaches infinity,

of the sum k goes from 1 to n minus 1 of a sub k is also equal to L.

Why is that?

Both of these amount of adding up a bunch of terms in the sequence and

seeing what I get close to.

Let me be a little bit more precise to see formally

why I can conclude that these two limits are both equal to L.

I mean assuming the series converges to L.

Let's see.

So, I can define the nth partial sum

as the sum k goes from 1 to n of a sub k.

And to say that the series, K goes from 1 to infinity

of a sub K has value L is just to say that the limit of

the nth partial sum as n goes to infinity is equal to L, right?

This is the definition of this.

Now, here's the subtle point.

The limit of Sn=L as n approaches infinity is the same as saying

that the limit of Sn-1 as n approaches infinity is equal to L.

Why is this the same as this?

Well this is saying that if I choose n big enough

I can get S sub n as close as I like to L.

Well this is really saying the same thing, all right?

If I choose n just a little bit bigger, one bigger,

then I can guarantee that S sub n minus 1 is just as close to L.

So asserting this limit is really the same as asserting this limit.

But this statement can be re-written using this.

So what this last statement is really saying,

is that the limit as n approaches infinity.

What's S of n-1?

Well, I'll use this up here.

That's saying the sum k goes from 1 to n-1 of a sub k = L.

So I've got two sequences, both of whose limits are L.

And that means the limit of their difference is zero.

Okay, so I've got these two sequences.

And they've both got a common limit of L.

So I'm going to take their difference.

Let's see what I get.

So if I take their difference, I get the limit as n goes to infinity

of the sum k goes from 1 to n of a sub k.

Minus the limit and goes infinity of

the sum K goes from 1 to n – 1 of a sub k.

And that’s L minus L.

So this difference is 0.

Now I can supply this a bit, all right.

This is a difference of limits which is the limit of the difference.

So this is the limit, as n goes to infinity of the difference of these two

things, which is the sum, k goes from 1 to n, a sub k minus

the sum, k goes from 1 to n minus 1 of a sub k.

But now, what's this?

Well this is really the limit as n goes to infinity of

this sum is a sub one plus a sub two plus dot, dot, dot,

plus a sub n minus, what am I subtracting here.

I'm subtracting a sub one plus a sub two plus dot, dot, dot plus A sub n-1.

So if I add up a sub 1 through a sub n, and then I subtract everything except for

a sub n, what I'm really taking the limit of is just a sub n.

So this is the limit as n goes to infinity of just a sub n by itself.

And what we've said here is that limit is 0.

So the limit of the nth term is 0.

So, what have we proved?

So the conclusion was that the limit of the nth term is 0.

But the assumption at the beginning was that the series converged to L.

So what we've really shown is the following, we've shown that if this series

converges then the limit of the nth term is 0.

So, let's turn that around.

Let's take the contrapositive.

So here's the original statement.

If the series converges then the limit of the nth term is 0.

And here's the contrapositive.

I just turned it around.

If the limit is not 0, meaning that either the limit doesn't exist or the limit

does exist but in some number that isn't 0, then the series has to diverge.

Because of the series did converge, then the limit would have to be zero.

What we have here is a test for divergence.

Well here's how this works.

This is the question that we are always being asked.

Question, does the series converge or diverge?

And what we can do now is we can take a look at the limit of the nth term.

And if that limit is not zero, then I know that the series diverges.

On the other hand, it's important to keep track of

the direction that this argument works in, right?

If it happens that the limit is equal to zero,

then the series might converge, it might diverge.

In that case, this test is silent.

It doesn't tell us any information.

But if we know that the limit is not 0, then I know that the series diverges.

Let's try this on an example.

So the original question was this.

Does the series, n goes from 1 to infinity

of n over n+1 converge or diverge.

We'll look at the limit of the nth term.

So let's look at the limit as n goes to infinity

of the nth term, which is n/n+1.

That limit is 1, and 1 is not 0.

So, the series n over n plus 1.

N goes from one to infinity diverges.

And that hopefully makes sense.

because to say that the limit of n over n plus one is equal to one means that this

series involves adding up numbers that eventually are very close to one.

And if you add up a bunch of numbers that are very close to one, well then you're

almost adding up one plus one plus one plus one and that certainly diverges.

In this case, because the limit isn't zero,

right, it can't be that the series converges.

Because if the series were to converge, then this limit would have to be zero, but

the limit's not zero.

So the series must diverge.

There's a lot of stuff going on here.

So if you're having some trouble keeping track of all the moving pieces,

here's a different way to think about what's going on.

Think way back to the beginning of this talk.

The very first thing we were looking at was this.

If a series converges, then the limit of the nth term is equal to zero.

Now, starting from that premise, we then concluded this,

that if the limit of the nth term is not zero, then the series diverges.

But putting these two statements next to each other,

they can be a little bit confusing, right?

Why is diverges the conclusion of this statement, but

converges is the assumption that I have to make over here?

Maybe they look like they're out of order.

Well, one way to think about this is to make it a bit more a real world.

Instead of thinking about series, let's think about rain and clouds.

If it's a rainy day, it's then a cloudy day, right.

The rain has to come from somewhere, so raining implies cloudy.

What happens if I negate this, right?

What happens if it's not raining?

Is it then necessarily not cloudy?

No, that's not true.

There's plenty of days when there's no rain but there's still clouds in the sky.

What if I turn this implication around?

Is this a true statement?

Yes.

If it's not cloudy, then it's not raining.

Because if it were raining it have to be cloudy.

So, if that same kind of thinking now that I want to apply to this

statement about series.

I'm starting with the statement that converges implies the limit of the nth

term is 0, all right.

I'm starting with a statement like rain implies clouds and

now I want to turn it around.

So I want to say not converges and not limit of the nth is 0.

But that means I also have to reverse the implication arrow.

This is called the contrapositive.

So what is this statement saying?

It's saying that if it's not the case that the lim an=0,

which I could write this way, then the series doesn't converge,

which is just another way to say it diverges.

And that's the statement that I ended with, right?

I'm ending with the statement that,

if the limit of the nth term isn't zero, then the series diverges.

But it's super important to keep track of the direction of this relationship.

This statement, diverges implies the limit of the nth term isn't zero.

That statement's not true, right?

Just like not rainy implies not cloudy isn't a true statement,

but this statement is true.

If the limit of the nth term isn't 0, then the series diverges.

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