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Absolute convergence?

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[MUSIC]

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As we've seen, practically all of the

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convergence tests that we have at our
disposal

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have made the assumption that the terms in

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the series are positive or at least
non-negative.

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I want to prove that the sum n goes from 1
to

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infinity of minus 1 to the nth power over
n squared, converges.

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So, how am I going to do this?

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I can't apply the usual convergence tests
because not all the terms

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are non-negative.

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I mean look, here I've written out some of
the terms, minus

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1 plus a fourth, minus a ninth plus a 16th
minus a 25th.

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I can only apply the comparison test if
the

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terms were non-negative, and that's not
the case here.

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But I can think about absolute
convergence.

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I can use the theorem that

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absolute convergence implies just regular
old convergence.

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So if I can just prove that this series
converges

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absolutely, then I know that it converges
in just the usual sense.

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So let's try that.

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What I know is this.

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The sum n goes from 1 to infinity of the

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absolute value of minus 1 to the n over n
squared.

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Well, that's exactly the same thing as the
sum n

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goes from 1 to infinity of just 1 over n
squared.

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And that's a p series, with p

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equals 2, and because 2 is bigger than 1,
this p series converges.

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Now, what does that mean then about the
original series I care about?

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That means that the sum n goes from 1 to
infinity of minus 1 to the n

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over n squared, converges absolutely
because some of the

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half loop values converges and
consequently, by the theorem,

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it just plain old, converges.
That's often how this is going to work.

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Let's suppose you want to analyze this
series, the sum

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n goes from 1 to infinity of a sub n.

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You've been given this task.

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Well, the first thing I'd suggest you do
is the limit test.

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Take a look at the limit of a sub n as n

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approaches infinity, because if that's not
0, then you know your series diverges.

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You're done.

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But let's suppose the series passes that
test,

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then what do you do?

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Well, then you can hope that the terms in
the series,

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the a sub n's are all greater than or
equal to 0.

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because if you've got a series, all of
whose terms

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are non-negative, then you can apply all
the usual convergence tests.

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But if that's not the case, if you're in a
situation where some of

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these terms are positive, some of these
terms are negative, what do you do?

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Well, then I'd recommend that you apply
all

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of our old convergence tests, not to this

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series directly, but to this series.

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The sum n goes from 1 to infinity of the
absolute value of the a sub n's.

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What I'm suggesting that you do is

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try to prove that this series converges
absolutely.

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Because if you know that this series
converges absolutely,

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then you know this series just plain old
converges.

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So, absolute convergence is an important
idea

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not because every single conversion series
converges absolutely.

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That's not even true.

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There are series that converge but don't
converge absolutely.

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Nevertheless, a lot of series do converge
absolutely.

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So an easy way to prove convergence, is to
prove absolute

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convergence and then to use the theorem
that absolute

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convergence just implies regular old
convergence.

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That's going to be successful a lot of the
time.

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[NOISE]