“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

级数

在这第二个模块中，我们将介绍第二个主要学习课题：级数。直观地说，将数列的项按照它们的顺序依次加起来就会得到“级数”。一个主要示例是“几何级数”，如二分之一、四分之一、八分之一、十六分之一，以此类推的和。在本课程的剩余部分我们将重点学习级数，因此如果你在有些地方感到疑惑，将会有大量时间来弄清楚。另外我还要提醒你，这个课题可能会令人感到相当抽象。如果你曾经为此困惑，我保证下一个模块提供的实例会让你感到豁然开朗。

- Jim Fowler, PhDProfessor

Mathematics

0.9 repeating.

[SOUND] Here's something that's often confusing,

0.9 repeating by which I mean 0.9999999,

going on like that forever, that is equal to one.

Now, not just close to one, not a little bit less than one,

this is exactly equal to one.

There's a few different ways to think about this.

For example, you might already believe that 0.3 repeating,

right, 0.3333 and so on is equal to one-third,

now if I multiply all of this by 3, what happens?

Well, if I multiply this by a 3, I get 0.9 repeating,

right, if I multiply 0.33333 by 3,

I get 0.99999 but if I multiply a third by 3, I get 1.

We can also formulate this in terms of a series.

I can instead think about it this way, 0.9 repeating is the sum

n goes from 1 to infinity of 9 times 10 to the -n.

Well why is that?

What's the first term here?

What happens when I plug in 1?

That's 9 times 10 to the 1st power,

when I plug in n=2 that's 9 times 10 to the -2nd power.

When I plug in n=3, that's 9 times 10 to the -3rd power and so on.

What are these?

All right, 9 times 10 to the -1 power is nine-tenths,

9 times 10 to the -2nd power is nine-hundredths,

9 times 10 to the -3rd power is nine-thousandths and it keeps on going.

Well, what's nine-tenths, that's 0.9.

What's nine-hundredths, that's 0.09.

What's nine-thousandths, that's 0.009 and it keeps on going.

And when I add these up, what I end up with is just a 0.9 from here,

this 9 gives me a 9 here, this 9 gives me a 9 here, the next term gives me the next

9 and so on, I end up with 0.9 repeating.

We can evaluate that series.

So the series is 9 times the sum n=1 to infinity of 10 to the -n,

and I can make that look even more like a geometric series.

It's 9 times the sum and goes from 1 to infinity of one-tenth to the nth power.

Now how do I evaluate that?

Look at a formula for summing an infinite series like that, it's a geometric series,

whose common ratio is between 0 and 1.

So that converges and its sum is 9 times the first

term is one-tenth divided by 1- one-tenth, what is that?

That's 9 times one-tenth divided by nine-tenths, well,

what's one-tenth over nine-tenths?

I could multiply the top and the bottom by 10 and

I get this is 9 times a ninth and 9 times a ninth is 1.

And this discussion brings up an important point,

anytime that we're writing down decimals, I mean if I just make up

some number like 0.57896 and imagine it keeps on going.

Any time I'm writing down real numbers like that,

I'm secretly writing down a series.

What do I even mean by I really mean a series,

I mean that this is the sum n goes from 1 to infinity of d sub n times 10 to the -n.

Where these d sub n's are the digits, right,

d sub 1 is 5, d sub 2 is 7, d sub 3 is 8 and so on.

Right, so anytime I'm writing down a decimal expansion,

I'm secretly writing down an infinite series just by adding up all the decimals.

To make it a little bit clearer,

let me just write down the first few terms, right, n = 1 and d sub 1 is 5,

means that this is 0.5 and to 0.5 I'm adding the n=2 term,

which is d sub 2 is 7 times 10 to the -2, which is 0.07.

And then I add the n = 3 term which is 0.008.

And then I'll add the n = 4 term, which is 0.0009 and so on.

So, all of these decimal representations of real numbers are secretly just

a series.

So, when thinking about real numbers or at least their decimal representations,

we're led naturally to think about series.

[SOUND]

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