Alternating Series

\textrm{We approximate the infinite series } \sum_{n=1}^{\infty}a_n \textrm{ by the truncated sum } \sum_{n=1}^N a_n. \\ \textrm{Here we demonstrate the alternating series test: A series } \sum a_n \textrm{ with either } \\ a_n = (-1)^nb_n \textrm{ or } a_n = (-1)^{n+1}b_n \textrm{ where } b_n \geq 0 \textrm{ for all } n. \textrm{ Then if, } \\ \textrm{ 1. } \lim_{n \rightarrow \infty} b_n = 0 \textrm{ and, } \\ \textrm{ 2. } (b_n) \textrm{ is a decreasing sequence } \\ \textrm{ the series } \sum a_n \textrm{ is convergent. } \\ \\ \textrm{Specifically, we show that the error term } | S(N) - L | \leq |a(N)| \textrm{, where we approximate the limit L by } S(1000).