We have seen earlier the wonderful news that all reasonable-looking alternating series converge. We have also seen highly worrying examples that show that some alternating series converge to different values depending on how the terms in the series are rearranged. This web page starts a short section on those nice series that converge in such a way that doesn't depend on any such rearrangement.
This definition, and its rather suggestive use of language begs an
important question: does a series that converges absolutely converge
at all? There is nothing in the definition that guarantees this, but
it wouldn't be so good if there was a series that converges absolutely
but doesn't converge. Fortunately for whomsoever first thought up the
definition, the answer is
yes. This is the main point of this
Absolute Convergence Theorem.
To prove this we should first recall the
Cauchy Property of convergent
sequences and restate it for the sequence of partial sums of a
series. A sequence (
and by the completeness of the reals,
Clearly, directly from the definition, any convergent series of positive terms converges absolutely.
Recall that our versions of the comparison or ratio
tests apply to series of positive terms. We now see
how to apply them to more general series
In other words, the comparison and ratio tests prove absolute convergence of series. You should show that you understand that the implication from absolute convergence to convergence is not a complete triviality by quoting it properly when needed.
For example, for any choice of
On the other hand, not every convergent series converges absolutely: