## Thomas' Calculus 13th Edition

$$10^{-5}$$
The Alternating Series Test states: Consider a series $\Sigma a_n$ such that $p_n=(-1)^n q_n$; $q_n \geq 0$ for all $n$ If the following conditions are satisfied then the series converges: a) $\lim\limits_{n \to \infty} q_n=0$; b) $q_n$ is a decreasing sequence. Consider $S_n=a_1-a_2+......+(-1)^{n+1}a_n$ and $$|S-S_n|=\leq |a_{n+1}| \\\implies |\space Error |=|S-S_4| \leq |a_{4+1}|=|a_5|\\=|(-1)^5 \times \dfrac{1}{10^5}| \\=10^{-5}$$