Alternating Series Test

Theorem (Alternating Series Test)

An alternating series of the form ${\sum }_{n=1}^{\infty }(-1{)}^{n+1}{b}_n$ or ${\sum }_{n=1}^{\infty }(-1{)}^n{b}_n$, where $b_n\ge 0$ for all positive integers $n$, converges if

i. $0\le b_{n+1}\le b_n$ for all $n\ge 1$, and
ii. ${\lim }_{n\to \infty }{b}_n=0$.