Does it converge?
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There's a somewhat standard process that you can use
to go about checking the convergence of a series.
You got a series.
The sum, n, goes from 1 to infinity a sub n.
And you want to know, does it converge?
I'd recommend applying the limit test first.
Because if you calculate the limit of a sub n as n goes to
infinity, and that's not 0, then you know the series diverges and you're done.
Then,
I try to check absolute convergence.
Right, So if this limits 0 then you don't
know, but you could try to investigate absolute convergence And
you could do that using any of the tests
that we have for series whose terms are not negative.
The root test. The ratio test.
The limit comparison test. What have you.
And if it converges absolutely, you're done.
But if it doesn't converge absolutely,
well, in that case you're back to just looking at the sum of the a sub n's again.
If your series has some positive and some negative terms and it
doesn't converge absolutely, well you better
hope that it's an alternating series.
Because if this is an alternating series, then at least you have the alternating
series test at your disposal. If it's not an alternating
series, well, you could try writing down
the sequence of partial sums, and try to
evaluate the limit just with your bare hands.
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