“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

What does the same mean?

[MUSIC]

Equality is a subtle topic.

The idea of four could just be conveyed by four dots,

it could also be conveyed by uppercase Roman numerals, lowercase Roman numerals,

or a ton of other symbols that have been used in various times and

places throughout the world and throughout history.

All of these mean the same thing, they all mean four.

They're all in some sense equal.

But that's just equality of symbols, of shapes.

It is something much more subtle happening for sequences.

When are two sequences the same?

Two sequences a and b are equal, they're the same, if they start at the same index,

which I'll call big N and corresponding terms are the same.

So that a sub n equals b sub n, whenever n is bigger than or

equal to that starting index.

Let's see how this works out in practice.

Here's one sequence, a sub n.

Here's the sequence that starts with a zero term and

is defined by the rule that its nth term is 2 to the n.

And here's another sequence b sub n.

The sequence b sub n whose 0 term is defined to be 1 and

subsequent terms will be calculated by referring back to previous terms so

the nth term is twice the preceding term.

These two sequences, a sub n and b sub n, they're the same, they're equal.

But they're written down really differently.

All right.

This sequence b sub n is defined recursively and

the sequence a sub n is just defined by a single formula in terms of n.

They both start with the term label zero, and

corresponding terms have the same value.

A sub 0 is 2 of the 0 using this formula which is 1,

and that's the same as b sub 0.

B sub 1 is using this recursive formula twice b sub 0.

B sub 0 is one so b sub 1 is two times 1 which is 2.

And that's the same as a sub 1 which is, using this formula, 2 to the 1st power.

And that pattern continues.

These two sequences both start with a zero term.

And each term of a sub n is twice the preceding term which is exactly

the recursive definition that I'm giving for b.

So a sub 1,000 equals b sub 1,000.

A sub a million equals b sub a million,

a sub anything equals b sub the corresponding thing.

So, the sequence a sub n and the sequence b sub n,

these two sequences are equal as sequences.

Equality isn't about outside appearances, it's what inside that matters.

The same for sequences.

Two sequences are equal, not if they've got the same outside form,

but if their corresponding terms have the same value.

[SOUND] [SOUND]

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