Find beginning of the asymptotic expansion of the sum

Question

Find beginning of the asymptotic expansion of the sum:

$$ (n!)^{-1}\sum^{n}_{k=1}k! $$

against the function $n^{-i}$, for $i\geq0$ to the nearest $\mathcal{O}(n^{-5})$.

Answer 1

Hint: Multiply $(n!)^{-1}$ into the sum. Prove that the result is an asymptotic expansion.