“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

数列

欢迎参加本课程！我是 Jim Fowler，非常高兴大家来参加我的课程。在这第一个模块中，我们将介绍第一个学习课题：数列。简单来说，数列是一串无穷尽的数字；由于数列是“永无止尽”的，因此仅列出几个项是远远不够的，我们通常给出一个规则或一个递归公式。关于数列，有许多有趣的问题。一个问题是我们的数列是否会特别接近某个数；这是数列极限背后的概念。

- Jim Fowler, PhDProfessor

Mathematics

Sequences are complicated.

[SOUND]

[MUSIC]

It's not so easy to find the limit of a sequence.

The logistic map provides one source of rich and

by rich I mean very complicated, examples.

Let's see how these are defined.

Here's a sequence I want to consider.

It's actually a family of sequences which depend on some parameter r.

Let's call r 2.5 for the time being.

And I'll pick an initial value for my sequence.

Let's make it 0.25 for now, and then here's the recursive

definition of the sequence, a sub n plus 1 will be this constant

r times the previous term a sub n, times 1 minus a sub n.

And by varying r and by varying the initial value,

I'll get a whole bunch of different sequences to consider.

Let's try it with those values.

Let's try it with r equals 2.5.

So here's an example.

I start with the initial value, 0.25.

And each subsequent value is 2.5 times the previous value times 1 minus

the previous value.

And here I've graphed these values, right.

The first is at 0.25 and then it goes up a bit and then it goes, and

then it starts sort of leveling off here.

And indeed if we look at this numerically, all right,

the first value is exactly 0.25.

And then as I run down through this sequence,

it seems like the values are getting closer and closer, to about 0.6.

And in this case the limit of this sequence is, in fact, 0.6.

What if we start with a different initial value?

What if instead of a sub 1 equals 0.25 we make a sub 1 equal to something else?

So the recursive formula we're using is a sub n plus one,

is 2.5 times a sub n times 1 minus a sub n.

And before we were starting with a sub 1 equal 0.25.

Let's start with 0.8 and see what happens in that case.

So if I start with 0.8, I could rewrite 0.8 as four-fifths and

then instead of writing 2.5, I'll write five-halves times an times 1 minus an.

Just so I can work with the fractions instead of the decimals.

So let's use this to calculate the next term.

What's a sub 2?

Well, according to this formula it's five-halves times a sub 1

times 1 minus a sub 1, but I've got a sub 1 that's four-fifths, so

that's five-halves times four-fifths times 1 minus four-fifths which is one-fifth.

Now one-fifth and this 5 can cancel.

I can get rid of these.

And this 4 and this 2, 2 in the numerator, so I've just got two-fifths.

So, a sub 2 is two-fifths.

Now I calculate a sub 3.

So, a sub 3 will be five-halves,

again times a sub 2, times 1 minus a sub 2.

And in this case, what is a sub 2?

We just calculated it, it's two-fifths.

So this is five-halves times two-fifths times 1 minus

two-fifths which is three-fifths.

But now I've got five-halves and two-fifths and

those cancel so a sub 3 is just three-fifths.

Now let's calculate a sub 4.

Well a sub 4 is again five-halves times a sub 3 times 1 minus a sub 3.

That's five-halves times a sub 3 is three-fifths.

One minus three-fifths is two-fifths.

But now I've got five-halves and two-fifths.

Those cancel so a sub 4 is just three-fifths.

Well what's a sub 5?

Well a sub 5 is also three-fifths.

I'm going to use the exact same formula here, but I'm again just going to plug in

three-fifths, and I'm going to get three-fifths out.

So a sub 5 is three-fifths, a sub 6 is three-fifths, right?

The point is that this sequence is just constant from here on out,

and that means that in this case the limit as n approaches

infinity of this sequence is just three-fifths.

Just like in the case when I started with 0.25,

right, happened in this case that even when I started with 0.8,

when I use this formula I again ended up with the same limit.

It's kind of interesting.

Lets try r equals seven-thirds.

I'll again have my sequence start with 0.25.

But then each new term is seven-thirds

times the old term times 1 minus the old term.

And if we look at this graph, it looks like this sequence is converging.

And if we look at the numbers, all right, the first term in the sequence is 0.25,

then it's about 0.4, then about 0.57 and it keeps on going.

And it does indeed look like this sequence is getting closer and closer to something.

This number might not be too meaningful to you, but

you can in fact show that the limit of this sequence is four-sevenths.

And indeed this number's about four-sevenths.

So changing the value of r seems to affect the limit.

Let's try r equals 3.25.

So again, my sequence will start with 0.25, but now the r value is 3.25.

What does the sequence look like?

Well I can graph a bunch of terms of this sequence, and it starts at a 0.25 and

it goes up, and then it seems to bounce between two values.

And, indeed, the numerical evidence supports that same conclusion.

Here is the first value, 0.25, that is about 0.6, 0.7, 0.5, 0.8,

0.5, 0.8, 0.5, 0.8, 0.5.

And it seems to just be flip-flopping between these two values.

And since it's flip-flopping between those two values, the limit doesn't exist.

Let's try another example.

Let's try r equals 3.7.

The same initial value 0.25 but now 3.7 is my value for r.

What does this sequence look like?

Well I can start graphing the terms in this sequence and

wow I mean it just looks like garbage.

There just doesn't seem to be any pattern at all.

And there's maybe moments when it looks like things are getting better but

then it suddenly breaks apart again.

And if you look at the numbers, right numerically things don't look so

great here either.

I mean, 0.25, that's the initial value but as you look through these numbers it

doesn't look like any sort of pattern is really coming out.

What is going on here?

Well, changing in the value of r doesn't just change

quantitative features of the sequence.

It changes qualitative features of the sequence.

Depending on the value of r, this sequence might converge, might have a limit.

Might flip flop between a couple values,

it might flip flop between four different values.

It might just move all over the place and

not really have any kind of discernible pattern.

And it all boils down to this value of r, in a seemingly mysterious way.

But that's not to say that the sequence can't be understood,

that it can't be studied.

All right, math isn't just random, right,

I mean there is a structure to this thing and with more work you can really start

digging into the very complicated structure that appears

even in something as seemingly simple as just a sequence of numbers.

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