Alternating series test

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

Let $(a_n{)}_{n=m}^{\mathrm{\infty }}$ 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 $\sum _{n=m}^{\mathrm{\infty }}(-1{)}^n{a}_n$ is convergent if and only if the sequence [...] converges to 0 as $n\to \mathrm{\infty }$.