Alternating series test

Alternating series test (Leibniz): Let $(a_n)$ be a sequence of real numbers such that

1. $a_n\ge 0$ for all $n$,
2. $(a_n)$ is decreasing, and
3. $a_n\to 0$.

Then the series $\sum_{n=1}^\infty (-1)^{n-1}a_n$ converges .

A common consequence is an error bound for partial sums : if $s$ is the limit and $s_N=\sum_{n=1}^N (-1)^{n-1}a_n$, then $|s-s_N|\le a_{N+1}$. This test is a basic source of conditional convergence and uses the necessary condition terms go to zero .