Answer
$S_{7}$ = $0.368056$
Work Step by Step
$S_{1}$ = $\frac{1}{0!}$ = $1$
$S_{2}$ = $\frac{1}{0!}-\frac{1}{1!}$ = $0$
$S_{3}$ = $\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}$ = $0.5$
$S_{4}$ = $\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}$ = $0.33333$
$S_{5}$ = $\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}$ = $0.375$
$S_{6}$ = $\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}$ = $0.366667$
$S_{7}$ = $\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}+\frac{1}{6!}$ = $0.368056$
$|S_{7}-e^{-1}|$ = $1.76\times(10)^{-4}$ $\lt$ $10^{-3}$
