“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

Why am I doing any of this?

[MUSIC]

Sequences are useful for a variety of reasons.

For starters, sequences help us understand repetitive processes.

And some of those repetitive processes are useful if we're trying

to compute something.

Well, here's an example of such a process.

I'll define a sequence recursively,

x sub n+1 will be 1 over x sub n plus x sub n over 2.

Maybe I'll start with the first term of the sequence as just being 1.

Let's start with x sub 1 equals 1, and see what we get.

x sub 2 is 1 over x sub 1,

which is 1, plus 1 over 2.

That's three-halves.

I can also try to calculate x sub 3.

I get that by taking 1 over x sub 2, so 1 over three-halves,

and adding that to three-halves over 2.

Now to do that calculation, well, I can write 1 over this fraction as two-thirds.

And instead of writing 3 over 2 divided by 2, I will write that as three-fourths.

Now I want to put this over a common denominator of 12.

So two-thirds is eight-twelfths, and three-fourths is nine-twelfths.

So altogether, x sub 3 is seventeen-twelfths.

We can compute more terms with the help of a computer.

Here's the x sub 2 term that we just calculated.

We can compute the next term, the x sub 3 term is seventeen-twelfths.

The x sub 4 term is 577/408.

Here's the x sub 5 term, and what you'll notice is that these

terms are getting closer and closer to the square root of 2.

Even x sub 3, which is seventeen-twelfths, is close to the square root of 2.

Let's see how that works.

So I want to try to convince you that seventeen-twelfths is

approximately the square root of 2.

Well, If I square both sides, what do I get?

I'm getting that 17 squared divided by 12 squared should be approximately 2.

And what's 17 squared?

Well, 17 squared is 289, and 12 squared is 144.

And is 289 over 144 close to 2?

Yeah, because the numerator is just about twice the denominator.

So one way to think about the square root of 2 is by thinking about this sequence.

If you want to find a really good approximation to the square root of 2,

all you've got to do is go far enough out, in this sequence.

[SOUND]

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