“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

Finding limits is hard.

I wish I could just prove that they exist.

[SOUND] Here's a theorem that

guarantees a sequence converges.

If the sequence is bounded and monotone, then the limit exists.

Why is this important?

Well, I often can't tell whether a sequence converges.

But I may be able to show that a sequence is both bounded and monotone.

And then I know it has a limit.

But why should I believe that the Monotone Convergence Theorem is true?

Well, let's think about this geometrically.

Suppose I've got a number line and I've got terms of my sequence,

x sub 1, x sub 2, x sub 3.

Let's pretend they are increasing, and

let's pretend that the sequence is bounded.

So, I know that the sequence never exceeds this value B.

So I've got a sequence which is increasing and bounded above.

Well, what can happen, right?

As I go out further and further in that sequence, I can never pass B.

And yet, I have to keep moving to the right.

So hopefully, it seems plausible that with these conditions,

this sequence can't help but converge to some limiting value L.

So hopefully, it seems plausible,

but we don't yet

have the tools to give a formal proof.

[SOUND] [SOUND]

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