“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

A list of numbers might keep getting bigger and bigger.

[SOUND] I want to precisely define

increasing for sequences.

A sequence, a sub n, is increasing If whenever m is bigger than n,

then the mth term is bigger than the nth term.

This is capturing the idea that the terms are getting bigger as I go further out

in the sequence.

Let's take a look at an example of an increasing sequence.

For example, the sequence a sub n equals n squared is an increasing sequence.

I can write out the first few terms.

The ace of one term is one.

It's 4, 9, 16, right?

These are just the perfect squares.

And the claim is this is an increasing sequence.

That as I go farther out in this sequence the terms get bigger.

And to really convince you of that formally, what I need to show is that

whenever m is bigger than n, then a sub m is bigger than a sub n.

And indeed, that's true, because a sub m is m squared, and a sub n is n squared.

And as long as these m, n are positive, this is true, whenever this is true.

We can also consider a sequence, which is decreasing.

A sequence a sub n is decreasing if whenever

m is bigger than n, then a sub m is less than a sub n.

This is capturing the idea that the terms are getting smaller.

In telling my that larger indexes correspond to smaller terms.

So as I go further out in the sequence terms are getting smaller.

It's easy to build examples of decreasing sequences.

If a sub n is an increasing sequence,

so an example of this would be the example we just saw.

An example would be a sub n = n squared.

Well if I've got an increasing sequence, then the sequence b sub n

defined by just negating the terms of a sub n is a decreasing sequence.

All right, if the terms of a sub n are getting larger,

the terms of b sub n are getting more negative, they're getting smaller.

Why even bother with all of this?

Well, basically I just want to give definitions for

the kinds of qualitative features that we might be interested.

We might be interested in a sequence that's increasing or

a sequence that's decreasing.

We might also be interested in a sequence which

maybe isn't necessarily getting bigger, but at least it isn't getting any smaller.

Is non-decreasing

if whenever M is bigger than N, then this isn't the case.

And to say that isn't the case means instead of less than, it's greater than or

equal to.

So a non decreasing sequence is not getting any smaller

in the sense that future terms are at least as large as previous terms.

Here's an example of a non-decreasing sequence.

This might start 1, 1, 2, 2, 3, 3, 4, 4, and so on where it

sort of repeats itself.

This sequence isn't increasing because later terms are not

larger than earlier terms.

2 is not greater than 2.

But the sequence is non-decreasing.

At least the terms aren't getting any smaller.

Not very surprisingly, we can also define non-increasing.

A sequence is non-increasing if,

whenever m is bigger than n, then a sub m isn't bigger than a sub n.

And isn't bigger means less than or equal to.

So a non increasing sequence is well not

necessarily decreasing but at least it's not getting any larger.

So thus far, we've talked about increasing and

decreasing, non-increasing and non-decreasing.

Generally, it's just interesting if a sequence is heading in the same direction,

if it's any of those things.

A sequence of a sub n is monotone, if the sequence is increasing or

non-increasing or decreasing or non-decreasing.

So monotone amounts to just a fancy way of saying heading in the same direction.

[SOUND]

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