“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

Let's think geometrically.

We're often exploring the limit of a sequence

by looking at some numeric evidence.

We might evaluate a few hundred terms in the sequence and notice how we're getting

really close to something numerically and then we might guess that that's the limit.

But we can also think about limits in a more geometric way.

So here I've got a number line and

I've plotted the terms in my sequence on that number line.

And it looks like the limit is L but what does that mean?

Well if I zoom in on the number line a bit,

it looks like all of the terms of my sequence are within

a hundredth of L as long as I'm after the fifty third term in my sequence.

And if I zoom in again all the terms of my sequence are within a thousandth of

L as long as I'm after the 181st term in my sequence.

And as close as I want to get to L I can do that as long as I go

far enough out in the sequence.

So, a limit's really a promise, and it's a promise that if

you want the terms of your sequence to be close to L, well you can do that.

You can get the terms to be as close as you want to L as long as you

throw away enough of the initial terms and

just restrict your attention to the tail of the sequence.

[NOISE]

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