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Absolute convergence.

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[SOUND].

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What do I even mean by absolute
convergence?

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Well here's the definition.

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Right, so defintion,

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the series, I'll write down a series, the
sum n goes from 1 to infinitiy of

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a sub n Converges absolutely.

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So this is the term that I'm defining.
Defining absolute conversion.

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So I'm going to say the series converges
absolutely if what happens?

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If the series,

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just the sum n goes from 1 to infinity of
the absolute values of the a sub n's just

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plain old converges in the usual sense.
What's an example of this?

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While ago, so that this series.

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The sum n goes from 1 to infinity of the
absolute

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value of sine n over 2 to the n; that
series converges.

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you could prove that by doing a comparison
test with the geometric

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series 1 over 2 to the n.

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Alright, so that series converges, and
that means that this

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series, the sum n goes from 1 to infinity
of

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just sine n over 2 to the n, no more

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absolute value now, just sine n over 2 to
the n.

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That series converges absolutely, or we
might say is absolutely convergent, just

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because this series, the sum of

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the absolute value of this quantity,
converges.

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But with a name

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like absolute convergence, you would hope.

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That an absolutely convergent series is
also

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just convergent in the regular sense of
converging.

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So that's a theorem.

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If the sum of the absolute values of the a
sub

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n's converges, then just the sum of the a
sub n's converges.

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In other words, if a series converges
absolutely then just plain old converges,

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let's prove that.
Okay yes so how's this proof going to go.

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Well the proof is going to start out

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with my assuming that the series converges
absolutely.

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Meaning I'm going to assume that the sum
of

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the absolute value of the a sub n's
converges.

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And then somehow you know, I want to do
something.

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I don't know what yet, so I've got some
cloud with question marks.

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And then, I eventually want to conclude
that the original series converges.

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So what goes in these question marks,
right?How

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do I get from this assumption to this
conclusion.

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Well the first thing I can note is that if
this

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series converges Then the sum of twice
those terms also converges.

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Why is that important?

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Well here is the key fact, this is sort of
a trick if you like.

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But what it's saying is that 0 is less
than or equal to a

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sub n plus the absolute value of a sub n
is less than or

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equal to twice the absolute value of a sub
n.

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Why is this even true?

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Well, think about the possibilities.
Maybe a sub n is negative.

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And if a sub n is negative, then I've got
a

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negative number plus well, the absolute
value of that negative number.

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These two things cancel, and I just get
zero.

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And then certainly this is true.

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Zero is less or equal to zero, is

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less than or equal to twice some positive
number.

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Well what if a sub n were positive?

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Well, then I'd have 0 is less than or

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equal to a positive number plus itself,
cause the absolute

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value of a positive number is just that
same

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number, is less than equal to twice that
same number.

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Well, in that case, these are equal right,
I've got a positive number plus itself.

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That's the same as twice that positive
number.

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So if a sub n's positive, this is true if
a sub

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n's negative, this is true if a sub n's 0,
this is

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just 0 less than or equal to 0 plus 0 less
than equal to 2 times 0.

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So whether a sub n is positive, whether
it's 0, whether

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it's negative, no matter what happens,
this is a true statement.

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Now we can apply the comparison test.

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So by the comparison test this series
converges right.

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The sum of these converges the side angles
from 1 to infinity

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of a sub n plus the absolute value of a
sub n.

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Well that's term wise less than this
convergent series and all

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the terms are non negative so that means
this series converges.

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but now our original series can be written
in terms of this

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series and the sum of the absolute values
in the a sub n's.

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So what I got is this series converges,
this

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series converges, but the difference of
these two series is

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the series that I'm originally interested
in, right.

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And the difference of convergent series
converges so that mean that

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the series just the sum of the a sub n's
converges.

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Which is exactly what I wanted to show.

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So we prove the theorem.
Alright, if a series converges absolutely,

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then it just plain old converges.
And that really justifies the terminology.

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I'm calling this, well everybody calls
this, absolute

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convergence, because it's a strong kind of
convergence.

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Absolute convergence implies convergence,
so absolute convergence

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is even stronger than just regular old
convergence.

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[SOUND]