Absolute convergence?
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As we've seen, practically all of the
convergence tests that we have at our disposal
have made the assumption that the terms in
the series are positive or at least non-negative.
I want to prove that the sum n goes from 1 to
infinity of minus 1 to the nth power over n squared, converges.
So, how am I going to do this?
I can't apply the usual convergence tests because not all the terms
are non-negative.
I mean look, here I've written out some of the terms, minus
1 plus a fourth, minus a ninth plus a 16th minus a 25th.
I can only apply the comparison test if the
terms were non-negative, and that's not the case here.
But I can think about absolute convergence.
I can use the theorem that
absolute convergence implies just regular old convergence.
So if I can just prove that this series converges
absolutely, then I know that it converges in just the usual sense.
So let's try that.
What I know is this.
The sum n goes from 1 to infinity of the
absolute value of minus 1 to the n over n squared.
Well, that's exactly the same thing as the sum n
goes from 1 to infinity of just 1 over n squared.
And that's a p series, with p
equals 2, and because 2 is bigger than 1, this p series converges.
Now, what does that mean then about the original series I care about?
That means that the sum n goes from 1 to infinity of minus 1 to the n
over n squared, converges absolutely because some of the
half loop values converges and consequently, by the theorem,
it just plain old, converges. That's often how this is going to work.
Let's suppose you want to analyze this series, the sum
n goes from 1 to infinity of a sub n.
You've been given this task.
Well, the first thing I'd suggest you do is the limit test.
Take a look at the limit of a sub n as n
approaches infinity, because if that's not 0, then you know your series diverges.
You're done.
But let's suppose the series passes that test,
then what do you do?
Well, then you can hope that the terms in the series,
the a sub n's are all greater than or equal to 0.
because if you've got a series, all of whose terms
are non-negative, then you can apply all the usual convergence tests.
But if that's not the case, if you're in a situation where some of
these terms are positive, some of these terms are negative, what do you do?
Well, then I'd recommend that you apply all
of our old convergence tests, not to this
series directly, but to this series.
The sum n goes from 1 to infinity of the absolute value of the a sub n's.
What I'm suggesting that you do is
try to prove that this series converges absolutely.
Because if you know that this series converges absolutely,
then you know this series just plain old converges.
So, absolute convergence is an important idea
not because every single conversion series converges absolutely.
That's not even true.
There are series that converge but don't converge absolutely.
Nevertheless, a lot of series do converge absolutely.
So an easy way to prove convergence, is to prove absolute
convergence and then to use the theorem that absolute
convergence just implies regular old convergence.
That's going to be successful a lot of the time.
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