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Monotonicity matters.

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[SOUND]

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Do you want

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to meet a statement of the alternating
series test?

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So here is what the alternating series
test tells us.

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If I've got a sequence, the terms of which
are decreasing, and they're all positive.

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And the limit of a sub n as n approaches
infinity is zero,

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then the alternating series that I build
using this sequence a sub n.

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The sum n goes from one to infinity of
negative one to the n plus one,

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times a sub n.
So this is an alternating series.

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This alternating series converges.

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I've been assuming that the sequence a sub
n was decreasing.

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And I used that to get that the even and
the odd partial sums were monotone.

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But what happens if I get rid of that
condition.

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What happens if I get rid of the condition
that a sub n is decreasing?

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Well, let's break the alternating series
test by removing

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this condition.

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I'm going to remove the condition that the
sequence a sub n be decreasing.

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I'll call this the Broken Alternating
Series Test.

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I'm just going to assume that the a sub

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n's are positive and that their limit is
zero.

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And if I just make these two assumptions,
is

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it then the case that this alternating
series necessarily converges?

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So to see how broken this really is,

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the question is this: can you think of an
alternating series

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where the terms are going to zero, but the
series diverges?

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Well, here's an example.

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It's the sequence a sub n.

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And the definition depends on whether n is
odd or even.

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So if n is odd, it's one over n plus one
divided by two.

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And if n is even, it's one over n over two
squared.

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Well, what does this look like?
I'm going to use this sequence

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to build an alternating series.

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And what is this alternating series, then
then, looking like?

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Well, I plug in n equals one, and that's
odd, so

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it's one over one plus one over two plus
one over one.

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I plug in n equals two.
I get one over two over two squared.

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But that's going to come with a minus
sign.

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I plug in n equals three.

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It'll be three is odd, so it's one over
three plus one over two, it's a half.

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I mean, here's some of the terms.
I just started

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writing them down.

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Alright, so it's one over one, and then
minus one over one

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squared, plus one over two minus one over
two squared, plus one

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over three minus one over three squared,
plus one over, I mean,

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what this really amounts to is sort of
weaving together two different series.

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Alright, the positive terms are the
harmonic series, and the

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negative terms in this alternating series
are the p series

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where p equals two.
And the limit of the nth term is zero.

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Well, specifically, the limit of the odd
terms here, it's

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just the limit of one over n, and that is
zero.

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And the limit of the even terms here, it's
just

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the limit of one over n squared, and
that's also zero.

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So putting together these two facts just
the limit of a sub n is equal to zero.

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But the sequence,

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it's not decreasing.

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If I look at the terms, all right, just

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the first few terms of this sequence a sub
n.

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I mean, these are not decreasing.
I mean, look.

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One half and then one fourth, but then one
third right?

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It's going up here, down here, up here,
down here, up here.

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It's not a decreasing sequence.
And the alternating series diverges.

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So I want to look at the sequence of
partial sums s and two n.

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The sum k goes from one to two n minus one
to the k plus one times a sub k.

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Now, the whole thing about this series is
that it's weaving

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together the harmonic series and a p
series with p equals two.

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So, this partial sum can be written like
this.

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The odd terms are the harmonic series.

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And the even terms are giving me this p
series, with p equals two.

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The odd terms are positive.
And the even terms are negative.

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So this partial sum is this.

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It's the sum k goes from one to n of one
over k

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minus the sum k goes from one to n of one
over k-squared.

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Now, the whole point here is that the
harmonic

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series diverges and this is a convergent p
series.

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So that means by taking n large enough, I
can make this harmonic series, or this

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partial sum for the harmonic series, as
large as I'd like.

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But no matter how big I take n, this
convergent p-series never

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exceeds the true value of this p-series,
which is pi squared over six.

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So what happens then, when n is really
large, right?

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This term becomes very, very large, as
large as I'd like.

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This term never exceeds the value of the
convergent p-series where p equals two.

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And that means

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that the limit of these even partial sums
is infinite.

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And consequently, this series diverges.

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because the limit of the partial sums
doesn't converge.

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This is a particular philosophy, by which
you can study the whole world.

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If you wouldn't know that car engines are
important.

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When you take the car engine out, then the
car doesn't go.

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So it must have been important.

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Well this same sort of game is being
played with the alternating

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series test.

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I took out a part of the alternating
series test.

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I've removed the assumption that a sub n
is decreasing.

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And I saw that the alternating series test
didn't work anymore.

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So although I don't emphasize it all the
time, the fact that the a sub

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n sequence is a decreasing sequence,
really is

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important when you're applying the
alternating series test.

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[SOUND].