“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

I want to be able to write down unending sequences of numbers.

[NOISE] If I want to give you a sequences

of numbers to analyze, how do I do that?

It’s not enough for me just to list off a finite number of terms,

I somehow have to provide you with all of the terms at once.

One thing I could do is give you a formula like a sub n equals n squared,

and this formula would define the sequence of perfect squares.

If I start with index 1 the sequence starts 1, 4, 9, 16, 25 and

so on, I can just give you the sequence by giving you this formula.

Sometimes, it can be quite hard to give a formula for the nth term of the sequence.

We could instead think recursively, what I mean is that I could give you

a formula that references previous terms in the sequence.

For example, we might begin the sequence with a 0 term that I'll define to be

equal to 0 and F sub 1 term which I'll define to be equal to 1.

And then I'll define future terms by referring to past terms.

So in this example F sub n will be defined as F sub

n-1 plus F sub n-2, and this is just an example.

Let's see how this works.

Let's say I want to compute F sub 2, how do I do that?

While I can use this formula but replace n with 2,

in that case I get F sub 2 is F sub 2 – 1 = F sub 2 – 2.

F sub 2 -1 is F sub 1 and

F sub 2 – 2 is F sub 0 and what's F sub 1?

Well I know these two facts, I know that F sub 1 is 1 and

I know that F sub 0 is 0 and consequently F sub 2 is 1 and we can keep going?

Let's compute F sub 3,

replacing n with 3 I find that F sub 3 is

F sub 3-1 plus F sub 3-2.

But F sub 3-1 is F sub 2 and F sub 3-2 is F sub 1.

Now I just computed F sub 2 a minute ago, F sub 2 is 1,

so that tells me that F sub 3 which is F sub 2 plus F sub 1,

is 1 plus F sub 1 is 1 and 1 + 1 is two,

so F sub 3 is 2 and we can keep going.

Well lets compute F sub 4, so if i want to compute F sub 4,

I replace n with 4, and I find that F sub 4 is F sub 4- 1,

plus F sub 4- 2,

4- 1 is 3 and

4- 2 is 2, so F sub 4 is F sub 3 + F sub 2, right?

Each term in the sum of the previous two terms.

Now, what's F sub 3?

Well, I just computed F sub 3 a moment ago, F sub 3 was F sub 2 + F sub 1,

which was 2, so, F sub 3 is 2.

And I computed F sub 2 a couple moments ago, and

I found that was 1, so F sub 4 is 3.

We're maybe beginning to see a pattern, 1, 2, 3 but that pattern does not continue.

Let's check F sub 5 is F sub 4 + F sub 3.

F sub 4 I just computed with 3 and

F sub 3 I computed a while ago with 2 and

3 + 2 is 5, so F sub 5 is 5.

What's F sub 6?

Well F sub 6 is F sub 5 plus

F sub 4, just computed F sub 5 and

it was 5 and F sub 4 was 3 and 5 + 3 = 8, so F sub 6 is 8.

This sequence has a name, so

this particular example is called the Fibonacci Sequence.

It starts 0, 1 and then each term is the sum of the previous two,

so the next term is 1, 0 + 1, the next term is 2, the next term is 3,

the next term is 5 the next term is 8, and so on.

This isn't to say that the only way

to talk about the Fibonacci Sequence is by giving a recursive formula.

We're going to see, later on in this course,

that it's possible to write down an explicit formula for

the nth Fibonacci Number but that's not really the main point here.

The main point is that, when you're thinking about sequences, or

you want to provide someone else with a sequence,

you don't have to give a formula just in terms of the index.

Your formula for the nth term can do more than just refer to n.

Indeed, a formula like this one can refer to previous terms in the sequence.

[SOUND]

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