Chapter 9 - Section 9.6 - Alternating Series and Conditional Convergence - Exercises - Page 521: 57

$ \approx 0.54030$

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