“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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From the course by The Ohio State University

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

Even nonsense, can be meaningful.

[SOUND] Mathematics is more than just things free of inconsistency,

it's more than just that which is the case.

Sometimes we're confronted with things that are really nonsensical, or

just things that are flat our wrong.

And when we're confronted with things like that, we should really have a feeling or

a need to salvage the situation.

We should take the nonsensical or the wrong thing and try to salvage it,

try to think of some sense in which it might make sense.

So what's the sum, n goes from 0 to infinity, of 9 times 10 to the nth power.

The party pooper simply says, the series diverges, and

the party pooper is right, this series diverges, and why does it diverge?

Well the geometric series with r = 10, and

that common ratio is bigger than 1 but

let's try to salvage the nonsense.

Remember, back, we looked at the sum n goes from 1 to infinity

of 9 times 10 to the -nth power?

And this was, it was equal to 1 but we can also write it as 0.9999 where

the 9s keep on going that way.

Well in this situation, contemplating, it needs to go to the opposite.

I'm thinking of the sum n goes from 0 to infinity of 9 times 10 to the n.

Well what is that?

When I plug in n = 0, I get 9 times 1 which is 9.

When I plug in n = 1, I get 9 times 10 which is 90.

When I plug in n = 2 I get 9 times 100 which is 900 and I'd keep on going.

So one way to write this might be, well at first, 9,

9 + 90, that's 99, 99 + 900, that's 999, it's

as if I could write it with nines going this way.

And what if we add 1 to that?

Well, I mean, it's not really a number, right?

But I can still write it down.

So I'll write down [LAUGH] A number that's just nines going this way and

then I want to add 1 to that number, what do I get?

Well, 9 + 1 is 10, carry the 1, 9 + 1 is 10, carry the 1,

9 + 1 is 10, carry the 1, 9 + 1 is 10, carry the 1,

9 + 1 is 10, carry the, and I'm going to keep on doing that, right?

So what it looks like this thing which is 9s all the way that way +1 is 0.

So whatever this thing is, it's a thing that if I add 1 to it, I get 0.

So, what is 999 with dots all that way?

It is sort of equal to -1, as -1 is a thing that I can add 1 to,

to get 0 and when I added 1 to this thing I got 0, sort of makes sense.

So if I just ignore convergence altogether, what would you tell me to do?

I'm trying to evaluate the sum, n goes from 0 to infinity,

of 9 times 10 to the n.

That would be 9 times the sum, n goes from 0 to infinity, of 10 to the n.

This is a, it's a divergent geometric series but

let's just pretend the formula still worked.

How do I calculate the value of that series?

That'd be 9 times, and the formula for this is, 1 over 1 -, the common ratio,

which is 10, of course that formula is only valid if it were a converging series,

and it's not, so let's just pretend.

What is this?

This is 9 times 1 over 1- 10, that's 9 times 1 over -9 and

9 times -1 over 9 is -1, which is sort of what were seeing here.

If we're just ignoring convergence,

it looks like it's telling us the value of this series.

It doesn't have a value because of divergence, but the value of the series,

maybe should be -1 and that's wrong because the series diverges.

But that's maybe the best of the wrong answers.

Well we can keep going with this.

For example, let's do some more calculations.

So let's start with this weird number which is nines all that way,

not really a number but there we go.

And let's multiply this by 5, what do we get?

5 times 9 is 45, so put the 5 down there and carry the 4,

5 times 9 is 45 plus 4 is 49, we got 4 to carry,

5 times 9 is 45 plus 4 is 49, got a 9 there and

I got to carry the 4, 5 times 9 is 45 plus 4 is 49, well, you get the idea.

I'm going to keep on getting 9s that way,

so ...9999 times 5 is 5999, well what is this?

Well what happens if I add 5 to it?

5 + 5 is zero, well it's 10 but I gotta carry the 1, 9 + 1 is 10,

so I put down a 0 and carry the 1, 9 + 1 is 10 so I put down the 0 and carry the 1.

So 5 with a bunch of 9s here + 5 is 0, so maybe then this number looks a lot to like

-5, because -5 is something I can add to 5 to get 0 and

we already saw that 9s this way looked a lot like -1.

So it seems like I've taken a number that's playing a role sort of like -1 and

I've multiplied it by 5 and I've got the number that's playing the role of -5,

which I could tell because when I added 5 to it, I got back to 0 and

that worked better than it should have.

What if we took 9999 and multiplied it by itself?

So I've got 9s all the way to the left, times itself,

9s all the way to the left, what is that product?

Well, 9 times 9 is 81, so we put the 1 down there and the 8 up there for

the carrying, 9 times 9 is 81, + 8 which is 89,

so put the 9 there, I got to carry this 8 now.

9 times 9 is 81 + 8 is 89, so I will write the 9 down there and I'm going

to keep on carrying 8s along the top and keep on writing 9s along the bottom.

So, it looks like working on the first digit here,

9 all the way to the left times 9, is just 1 with 9's all the way to the left.

But that's just working on this first digit, now,

I've gotta move to the next digit on the bottom.

So, put a 0 there, and it will 9 times 9, and it will 81, so,

I put the 8 up there, and the 1 down there, 9 times 9 + 8 is 89, so

put the 9 down there and then the 8 up there, 9 times 9 is 81 + 8 which is 89,

so I'll put the 9 down there and I'll put the 8 up there.

And I'll keep on going the same exact way, right.

So working on that second digit I've got a 0, a 1 and then all 9s.

Now I gotta work on the third digit on the bottom.

I put down two 0s here, and it's exactly the same pattern,

it's going to be 1, followed by 9s going on forever.

And then to work on the next digit on the bottom, it'll be three 0s and

then the same pattern of 1 followed by a bunch of 9s and

it's going to keep on going like that.

Well, after I do all the digits along the bottom, and it's all done,

I'm supposed to add them all up.

So now I've got 1, 9+1 is 10, so I put the 0 there

and I carry the 1, 1+9+9+1=12 so that's 0 here and I gotta carry a 2 there,

2+9+9+9+1, well that's 30 so I put the 0 there and the 3 here.

And I'm going to keep on going like that and

I'm going to get 0s all the way across here, so what just happened?

Well, it looks like we started with a number that's playing the role of -1.

And we multiplied it by a number that's playing the role of -1 and

when I actually did that multiplication, the answer that I ended up getting

to be 1, which is just what I'd hope it would be.

So it really is seeming like it's working and

we started out with this nonsensical thing, right, this divergent series but

we're sort of taking the nonsense seriously.

We've ended up getting a system that works sort of like the negative numbers.

And in fact, computers do store negative numbers this way.

Of course computers usually don't work base 10, they usually work base 2 and

we work base 2 into same kind of game and call it Two's Complement Arithmetic.

And this sort of thing comes up in just pure mathematics.

This sort of thing is often called the p-adic numbers, and

in particular when we're using powers of 10, 10-adic numbers.

So mathematics is so powerful, that even nonsense can lead to a reasonable theory.

It's one thing to win battles just by force alone, right,

to be great by virtue or simply being correct all the time.

But it is something else entirely to win those battles when you're weak,

to be on the right track even when you are entirely wrong.

And I think the fact that mathematics works like this really shows that there's

something to it.

It's not just symbols that we're pushing around on a page, we're really out there,

exploring something that's really there, something really beautiful.

And we're just fortunate to be able to get to take part.

[NOISE] [SOUND]

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