Evaluating series with factorial denominator (sanity check).

Your solution is not only correct but also a very nice one.

Once you know the closed form for the partial sum, you can also prove it by induction, starting with $S_1=1-1/2=1/2$ and taking the induction step

$$ \begin{align} S_{m+1}&=S_m+\frac{m+1}{(m+2)!} \\ &=1-\frac1{(m+1)!}+\frac{m+1}{(m+2)!} \\ &=1+\frac{m+1-(m+2)}{(m+2)!} \\ &=1-\frac1{(m+2)!}\;. \end{align} $$