“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

交错级数

在第四个模块中，我们讲解绝对和条件收敛、交错级数和交错级数审敛法，以及极限比较审敛法。简而言之，此模块分析含有一些负项和一些正项的级数的收敛性。截至目前为止，我们已经分析了含有非负项的级数；如果项非负，确定敛散性会更为简单，因此在本模块中，分析同时含有负项和正项的级数，肯定会带来一些新的难题。从某种意义上，此模块是“它是否收敛？”的终结。在最后两个模块中，我们将讲解幂级数和泰勒级数。这最后两个课题将让我们离开仅仅是敛散性的问题，因此如果你渴望新知识，请继续学习！

- Jim Fowler, PhDProfessor

Mathematics

Alternating series are awesome.

[MUSIC]

Why do we care about alternating series?

Well, if you're trying to analyze the convergence of a series and

all of the terms in the series are non-negative,

then you can break out all your usual convergence tests, the comparison test,

the root test, the ratio test, the integral test.

What about series where not all the terms are non-negative?

So maybe I'm trying to analyze a series where this doesn't happen,

it's not the case that all the terms are non-negative, but

I've got some positive terms and some negative terms in my series.

Well then what am I supposed to do?

Well one thing I could do in this case is instead of analyzing this series directly,

I could look at this series.

The sum n goes from 1 to infinity of the absolute values of the a sub n's, try to

show that this series converges absolutely and therefore, this series would converge.

And what about series that don't converge absolutely?

Let's suppose I want to analyze this series,

the sum, little n goes from 1 to infinity, -1 to the n + 1 divided by n.

Now, my first inclination would be to take a look at this series,

the sum of the absolute values of these terms,

with the hopes that I could maybe prove that this thing converges absolutely.

But that's not true,

this series doesn't converge absolutely because this series diverges.

What is the absolute value of -1 to the n + 1 over n?

This [LAUGH] It's just 1 over n, this is the harmonic series,

the harmonic series diverges, so this series does not converge absolutely.

So we've gotta do something else and

indeed, the ultimate in series tests comes to save the day.

I'll rewrite this series like this, as the sum n goes from 1 to infinity of -1 to

the n+1 times a sub n, where a sub n is 1 over n.

And now what I know is that the sequence a sub n is a decreasing sequence,

all of the terms of that sequence are positive and the limit of a sub n is 0.

It's the limit of 1 over n as it approaches infinity is 0.

And that means by the alternating series test,

the series that I'm studying here converges.

Now I've just shown that it doesn't converge absolutely.

So what the alternating series test is actually showing is that this series

converges conditionally, this is partly why alternating series are so important.

Well because of the alternating series test, we can prove that an alternating

series converges without using our other convergence tests

on the series of the absolute values, without proving absolute convergence.

We don't honestly have that many other tools for showing that a series,

some of whose terms are positive, some of whose terms are negative converges at all.

Normally, the way that we'd approach those is by showing that they converge

absolutely.

So, what are we suppose to do about those series which don't converge absolutely but

do converge?

And what can we do about the conditionally convergence series in our world?

Well the alternating series test is a great tool in our toolbox and

the alternating series test gives us more than just convergence.

Suppose I want to approximate the value of this series, well what could I do?

It's an alternating series, so I know that the even partial sums, in this case

will be underestimates and the odd partial sums will be overestimates.

So here's one of those odd partial sums,

the sum of the first three terms, 1 over 1- 1 over 2 + 1 over 3,

that's an overestimate of the true value of this series.

And here's an even partial sum, sum of the first 4 terms and

that's an underestimate in the true value of this series.

And then I could actually figure out what these are,

1- a half is a half + a third, that's five-sixths.

And here I've got five-sixths- a quarter, that's seven-twelfths.

So I know that seven-twelfths is less than or

equal to the true value of this series, is less than or equal to five-sixths.

Let me rewrite five-sixths, right I could call five-sixths ten-twelfths,

makes it a little clearer I think it’s actually bigger than seven-twelfths.

Now it turns out that these series of value is actually log 2, so

what I’ve done here is shown that log 2 is between seven-twelfths and ten-twelfths.

What I've really done is I've shown this, I've got this inequality now but this

is just coming because I've expressed log 2 as the value of an alternating series.

An alternating series provide this convenient error bounds on my estimates.

I could multiply everything here by 12 and that would tell me that 7 is less than or

equal to 12 times log 2, which by properties of logarithms

is the log of 2 to the 12 and that's less than or equal to 10.

And then I could e to the all three of these things, right,

I could apply the exponential function to all three of these things and

I'd find out that e to the 7th is less than or equal to 2 to the 12th cause

e to the log undoes the log, is less than or equal to e to the 10th.

And 2 to the 12th is 4096, so what I've shown,

just by playing around with alternating series is this,

that 4096 is between these two numbers.

And I mean this isn't great, these bounds aren't fantastic,

I didn't add up very many terms in this series, right.

But still I think it demonstrates the principle that one of the coolest things

about alternating series,

is that alternating series provides these convenient bounds for you.

[SOUND]

[MUSIC]

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