“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。

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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。 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

I want to know where a sequence is heading.

[SOUND]

[MUSIC]

Lets think about a specific sequence.

Here's an example to think about.

The sequence of a sub n defined by this formula,

the numerator is 6n + 2 and the denominator is 3n + 3.

We can list off the first few terms.

For instance, I want to compute a sub 1, I just plug in 1 for n.

I compute 6 x 1 + 2 divided by 3 x 1 + 3.

That numerator is 8, that denominator is 6.

And I could rewrite eight-sixths as four-thirds in lowest terms.

And I could keep on going.

By plugging in 2, I find that a sub 2 is fourteen-ninths,

a sub 3 is five-thirds, a sub 4 is twenty-six-fifteenths,

a sub 5 is sixteen-ninths, and I could keep on going.

Those fractions really aren't providing with much insight.

What's the eventual value of the sequence, where is the sequence heading?

Make it a better sense by plugging in a thousand say,

what's the thousand terms or that's 6 times

1,000 + 2 over 3 * 1000 + 3.

That's 6,002 over 3,003.

Well, that's awfully close to 2.

I can go even farther out in the sequence.

a sub 1,000,000 Is what?

It's 6 times a million plus 2 divided

by 3 times a million plus 3 and

that works out to be 6 million and

2 divided by 3000003.

And that's insanely close to 2.

All right.

The limit of a sub n as n approaches infinity is 2 to express this idea.

Okay, so what does that really mean.

Well, I promise that a sub n is as close as you want to 2,

provided that n is large enough.

For better or for worse, these things are usually written out with fewer words and

more symbols.

Instead of saying, as close as you want to 2, I'll say that a sub n

is within epsilon, some small positive number, of 2.

And how large is large enough?

Well, instead of saying large enough,

I'll just say that n is at least as big as sum index big N.

So I can say it like this.

So, for every epsilon greater than 0,

there is an index big N so that whenever little n is bigger than or

equal to big N a sub n is within epsilon of 2.

The idea here is that this epsilon is measuring how close you want

a sub n to be to 2, and I'm telling you that no matter how close you

want a sub n to be to 2, if you go out far enough in the sequence,

all the terms after that are actually that close.

Instead of writing with the epsilon of two you’ll normally it with absolute value.

So instead of saying within epsilon the two, I'll say

that the distance between a sub n and two is less than epsilon.

Put it all together, to say that, the limit of a sub n

as n approaches infinity equals 2, means for every epsilon greater than 0,

no matter how close you want to be to the limiting value of 2, there's some Index N,

so that whenever you're farther out in the sequence than big N, whenever little n is

bigger than or equal to big N, then the absolute value of a sub n minus 2,

which is measuring the distance between a sub n and 2,

that absolute value is less than epsilon.

Of course, there's nothing really special about the 2 in this.

So in general, to say that the limit of a sub n = L means that for

every epsilon there is a whole number big N, so

that when ever n bigger than or equal to N,

the absolute value of a sub n minus that limiting value L is less than epsilon.

[SOUND]

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