“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

What are sequences?

[MUSIC]

A sequence is a list of numbers.

For example, here's a sequence,

which is a sequence that goes 1, 1, 2, 3, 5, 8.

And I'll write dot, dot, dot, to remind that the sequence goes on forever.

It's an unending list of numbers.

Each of these numbers in the list is referred to as a term in the sequence.

I want some notation so I can refer to specific numbers in the list.

Well, I'll write a sub 1 for the first term in my sequence, a sub 2 for

the next term, a sub 3 for the next term, a sub 4 for the next term,

a sub 5 for the next term, a sub 6 for the next term and so on.

So in this particular example, I could say that the sixth term in

my sequence is 8 or the third term in my sequence is 2.

If I want to talk about the sequence as a whole,

I'll perhaps just write down (a sub n), maybe in parentheses.

And I'll use this notation to talk about the entire sequence.

But by plugging in different values for n,

I can then speak of specific terms in the sequence.

This means that a sequence really amounts to some assignment from these indices,

the subscripts to other real numbers.

And a gadget that assigns numbers to other numbers, it's a function.

So yeah, the sequence a sub n is secretly a function f(n).

Because both of these gadgets are just ways of assigning real numbers to

other numbers.

In this case, the sequence assigns a real number to these ns, to 1, 2, 3, and so on,

so it's worth pointing out then that what's really the domain

of this function that's playing the role of the sequence?

Well the domain of f should just be whole numbers.

I don't want to talk About the five-thirds term of the sequence, right.

It doesn't really make sense, usually, to talk about a sub five-thirds, or a sub pi.

But it does make sense to talk about a sub 3 and a sub 100.

So the domain of my function, the thing that I'm allowed to plug in for

n, should really just be whole numbers.

And I'll usually denote that with this fancy looking N.

I can take a look at a sequence numerically.

I can just take a look at the terms and see what their approximate values are.

Besides thinking numerically, I could also think about a sequence geometrically.

For example, here's a number line.

Say here's 0, here's 1, here's 2, here's 3 and it keep on going,

and I could try to plot the terms of my sequence on this number line.

And you could imagine some sequence where the first term is here between 0 and

1, the second term is between 1 and 2, maybe the third term is here between 1 and

the second term, maybe the fourth term is over here between a sub 1, and 1.

Maybe the fifth term is over here just a little bit less than 3.

You can plot the terms and

your sequence to try to get some idea about what the sequence look like.

I can think about a sequence algebraically.

Well, algebraically,

I could define a sequence by giving you a rule to compute the nth term.

Maybe a sub n is some algebraic formula, like n squared plus 1.

And in that case, I can just compute the first term is 1 squared + 1 = 2.

The second term is 2 squared + 1 = 5.

The third term is 3 squared + 1 = 10, and so on.

Our goal now, is to build some interesting sequences, and

explore the properties that those sequences have.

[SOUND]

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