Chapter 10: Infinite Sequences and Series - Section 10.6 - Alternating Series and Conditional Convergence - Exercises 10.6 - Page 603: 57

$\approx 0.54030$

Consider $S_n=a_1+a_2+......+(-1)^{n+1}a_n$ $\implies |S-S_n| \leq |a_{n+1}| $ and $\space |Error| \lt |a_{n+1} |$ $\implies |\dfrac{1}{(2n)!}| \lt \dfrac{5}{10^6}$ $\implies (2n)! \gt \dfrac{10^6}{5}$ $ \implies n \ge 5$ So, we need at least 5 terms: $1-\dfrac{1}{2!}+\dfrac{1}{4!}-.... \approx 0.54030$