“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。

Loading...

From the course by The Ohio State University

微积分二: 数列与级数 (中文版)

46 ratings

“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

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

Let's recycle sequences.

[MUSIC]

If I've got a sequence in hand,

I can build a new sequence out of that old sequence.

Well, here's a sequence to start with, it's a sequence,

the first three terms of which are all 1.

So a sub 0, a sub 1, a sub 2 are all equal to 1.

And that it compute each subsequent term, I just add up the three previous terms.

Let's see some terms, here are some terms in this sequence,

first three terms are all 1, 1, 1, 1.

A sub 0, a sub 1, a sub 2,

each subsequent term is computed using this recursive formula.

Well this formula tells me to use to add up the previous three terms, so

this 3 is coming from 1 + 1 + 1, this 5 is coming from 3 + 1 + 1,

this 9 is coming 5 + 3 + 1, this 17 is coming from 9 + 5 + 3, and so on.

Now this might seem a little bit like the Fibonacci Sequence except instead of

adding up the previous two terms I'm adding up the previous three terms.

So to be a little bit funny people sometimes call this the Tribonacci

sequence.

I can build a new sequence out of the Tribonacci sequence.

So starting with this sequence I could build a new sequence,

a sequence I'll call b by referring to the terms in this Tribonacci Sequence.

Let's see some terms of the sequence b.

Here is the sequence a sub n, all right it goes 1,1,1, 3, 5,

this is the Tribonacci sequence.

And here is the beginning of the sequence b sub n.

According to this rule,

b sub n is just the ratio of neighboring terms in the tribonacci sequence.

So for example, this term here, which is b sub 0, b sub 1,

b sub 2, b sub 3, is the ratio of a sub 4 and a sub 3.

Where this term here, which is b sub 4, is the ratio of a sub 5 and a sub 4.

The fractions are sort of confusing, so you can replace these

fractions with decimal approximations instead of looking at these fractions.

Here are some of the terms of b sub n, but written out in terms of

their decimal approximations, let's look at these approximations.

Remember, these are decimal approximations to neighboring

terms in the Tribonacci sequence and as we go out further and

further in the Tribonacci sequence, do you notice something?

Well, in light of this numeric evidence, it certainly looks like this sequence,

b sub n has a limit and this limit is about 1.839, so that raises a question.

So that's a puzzle for you, yeah, this limit turns out to exist and

it's about 1.8, but what is it exactly?

What is this limit exactly equal to?

Can you find a formula for this limit?

That turns out to be an extremely challenging question but

I hope it wets your appetite.

With not very many tools at hand, it's possible to build relatively

simple seeming things that are actually quite complicated to describe precisely.

[NOISE]

Coursera provides universal access to the world’s best education,
partnering with top universities and organizations to offer courses online.