\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \)

Alternating series test

An alternating series converges if the sequence of elements converges absolutely to zero.

Let \( (a_n)_{n=m}^{\infty} \) be a sequence of real numbers which are non-negative and decreasing, thus \( a_n \ge 0 \) and \( a_n \ge a_{n+1} \) for every \( n \ge m \). Then the series \( \sum_{n=m}^{\infty}(-1)^n a_n \) is convergent if and only if the sequence [...] converges to 0 as \( n \rightarrow \infty \).