Chapter 10: Infinite Sequences and Series - Section 10.6 - Alternating Series and Conditional Convergence - Exercises 10.6 - Page 603: 58

$\approx 0.3678819444$

Consider $S_n=a_1+a_2+......+(-1)^{n+1}a_n$ and $|S-S_n| \leq |a_{n+1}|$ and $ |Error| \lt |a_{n+1} |$ Now, $|\dfrac{1}{a!}| \lt \dfrac{5}{10^6} \implies a \ge 9$ So, we need at least $9$ terms. Thus, $S \approx S_8 =1-1+\dfrac{1}{2!}-\dfrac{1}{3!}+.... \approx 0.3678819444$