Editing Alternating series (section)

== Absolute convergence ==
A series <math display=inline>\sum a_n</math> [[absolute convergence|converges absolutely]] if the series <math display=inline>\sum |a_n|</math> converges.

Theorem: Absolutely convergent series are convergent.

Proof: Suppose <math display=inline>\sum a_n</math> is absolutely convergent. Then, <math display=inline>\sum |a_n|</math> is convergent and it follows that <math display=inline>\sum 2|a_n|</math> converges as well. Since <math display=inline> 0 \leq a_n + |a_n| \leq 2|a_n|</math>, the series <math display=inline>\sum (a_n + |a_n|)</math> converges by the [[Direct comparison test|comparison test]]. Therefore, the series <math display=inline>\sum a_n</math> converges as the difference of two convergent series <math display=inline>\sum a_n = \sum (a_n + |a_n|) - \sum |a_n|</math>.