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Does it converge?

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00:00:07,990 --> 00:00:10,390
There's a somewhat standard process that
you can use

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to go about checking the convergence of a
series.

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You got a series.

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The sum, n, goes from 1 to infinity a sub
n.

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And you want to know, does it converge?

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I'd recommend applying the limit test
first.

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Because if you calculate the limit of a
sub n as n goes to

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infinity, and that's not 0, then you know
the series diverges and you're done.

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Then,

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I try to check absolute convergence.

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Right, So if this limits 0 then you don't

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know, but you could try to investigate
absolute convergence And

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you could do that using any of the tests

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that we have for series whose terms are
not negative.

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The root test.
The ratio test.

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The limit comparison test.
What have you.

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And if it converges absolutely, you're
done.

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But if it doesn't converge absolutely,

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well, in that case you're back to just
looking at the sum of the a sub n's again.

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If your series has some positive and some
negative terms and it

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doesn't converge absolutely, well you
better

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hope that it's an alternating series.

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Because if this is an alternating series,
then at least you have the alternating

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series test at your disposal.
If it's not an alternating

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series, well, you could try writing down

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the sequence of partial sums, and try to

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evaluate the limit just with your bare
hands.

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