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Let's rearrange.

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[MUSIC]

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The alternating harmonic series is a

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great example of a conditionally
convergent series.

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Well, here's the alternating harmonic
series.

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It's the sum n goes from 1 to infinity of
negative 1 to the n plus 1 over n.

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And what we know about this is that it

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converges by the alternating series test,
but it doesn't

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converge absolutely, cause if you look at
the sum

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of the absolute values, you're looking at
the harmonic

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series, which diverges.

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And because it converges but not

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absolutely, we call it conditionally
convergent.

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And what about the negative terms?
What if I just add up the negative terms?

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Well, here's the first dozen or so terms
of the alternating harmonic series.

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So 1 over 1 minus a half, plus 1 3rd minus
1 4th plus 1 5th minus 1 6th and so on.

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What if I just add up these negative
terms?

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Just the terms with even index?
What do I get?

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Well, in that case, what I'm looking at.

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Is the sum n goes from 1 to infinity of 1
over 2 n.

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And these are the even index terms from
the alternating harmonic series.

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What do I get if I add up all of these
numbers?

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Well that's one half of what I get when I
add up 1 over n goes from n to infinity.

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But, that's half of the harmonic series.
Right?

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That means that this diverges.

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What if I just look at the positive terms?

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Well in that case, try to add up 1 over 1,

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plus a 3rd, plus a 5th, plus a 7th, plus a
9th.

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Alright, I'm trying to figure out just how
big is this?

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What's the sum n goes from 1 to infinity
of the 1 over odd numbers?

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Right, 1 over 2n minus 1.
Does this

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converge or diverge?

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Well the trouble is that 1 over 2n minus 1
is even bigger than 1 over 2n.

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Right, 1 over 1 is bigger than a half, a
3rd is bigger than a

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4th, a 5th is bigger than a 6th, the 7th
is bigger than a 8th.

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So if this series diverges, then this
series diverges as well.

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The sum of just the positive terms in the
alternating harmonic series diverges.

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And yet smoehow that series Has a finite
value, so in the alternating

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harmonic series, the negative terms
diverge, the positive terms diverge.

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They both diverge.

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So what I've got is really two piles of
numbers, and if I take enough

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from either pile, I can make a number
that's as large as I would like.

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This presents me with the following

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quite strange opportunity.
Here's my goal.

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I'm going to rearrange the terms of the

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alternating harmonic series to get a new
series.

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Same terms, just different order, but now
my new

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series, when I evaluate it, will have
value 17.

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I'll keep picking up positive terms until
I exceed 17.

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Okay, well here we go.

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Here's the number line, here's zero,
here's my goal, 17.

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Trying to pick

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up numbers from this pile so that I can
move all the way past 17.

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So I could just get started, right, I pick
up one, and that gets me

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a little bit closer to 17, I'll pick up
the next number, here's a 3rd.

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That gets me a bit closer to 17.
I'll pick up a 5th.

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That gets me a little bit closer to 17.

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I'll pick up a 7th, right, and that gets
me even a little bit closer to 17.

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I mean, the trouble, of course, is that
these numbers

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are getting smaller, but I know that this
series diverges.

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So if I keep taking numbers from this
pile, I

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can move as far to the right as I like.

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And indeed, it happens that the sum n goes
from 1 to 10 to

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the 14th of 1 over 2n minus 1 is a bit
bigger than 17.

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It's 17.1, so this is how I'm going to
start.

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I'm trying to write down the same terms as
the alternating harmonic series,

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but I want a series now that converges to
17.

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And this is how I'll start, I'll add 1
over 1 plus a 3rd plus

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a 5th all the way to 1 over 2 times 10 to
the 14th minus 1.

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And that'll land me just to the right of
17.

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And now I'll use some of the negative
terms.

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The sum of these terms from this positive
pile was just about 17.1 So

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if I pick up a half from the negative
pile, and I'm

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going to subtract a half now, that moves
me over to about 16.6.

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Now I'll take some of the positive terms
again.

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Of course I've already used up a lot of
positive terms, but

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there's definitely more there that I can
add, because this series diverges.

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I've only taken away a finite piece of it,
so there's more yet to grab.

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It's just

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the numbers are, are really big.
Or really small, rather.

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But in any case, there's more terms from
this positive pile to add,

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and if I add enough of them, I'll
eventually move past 17 again.

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Maybe I'll end up at, say, 17.001, or
thereabouts.

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And some more of the negative ones.

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I'll pick away this quarter, I'll subtract
a

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quarter from here, and now my 17.001 or
thereabouts

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may be moves over to a little bit less
than 17, say 16.751.

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And I'll just keep on doing this.
I can add more positive terms again.

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And move myself back to the other side of
of 17, and

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I'm never going to run out, right, cause
this pile of numbers is infinite.

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I mean, this series diverges.

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So I'm going to keep moving back and forth
past 17.

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And in the limit, I'll get 17.

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So what I mean is that by adding up the
same terms.

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In the alternating harmonic series, just
in a different order.

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I'm able to write down a series whose
value is 17.

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But does that mean that the value of the
alternating harmonic series is 17?

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No.

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No.

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We're eventually going to see that the

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true value of the alternating harmonic
series

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Is log 2.

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What we're seeing here is the first
glimpse of a theorem.

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It's a rearrangement theorem.

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Here's how it goes.
Suppose that L is some real number.

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You get to pick out.

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And you've got a conditionally convergent
series.

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In this case, I'm calling it the sum n
goes from 1 to infinity of a sub n.

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So L's a real number that you picked.

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And you're given this conditionally
convergent series.

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Then that sequence

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a sub n can be rearranged to form a new
sequence, b sub n.

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So the b sub n sequence contains all the
terms

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of the a sub n sequence, just in a
different order.

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Well that rearragned sequence b sub n, if
you form a series out

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of it, the sum n goes from 1 to infinity
of b sub n.

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The value of that series is L, and you
picked L.

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What this is saying is that if you're

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given a conditionally conversion series,
you can rearrange

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the terms so that that series sums to any
number that you'd like.

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In light of this theorem, we have to be
careful about how we think about series.

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Order matters.

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A series is a list of numbers to sum in a
given order.

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It's not just a pile of numbers that you
add up.

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The numbers are coming at you in a given
order.

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[NOISE]