Denote the number of derangements by $D_N$. It's known that $D_N/N! \rightarrow 1/e$. Therefore $N!/e$ is an approximation for $D_N$.

I'm trying to bound the difference between this approximation and $D_N$. Namely, to bound from above $|N!/e - D_N|$ as a function of $N$.

I believe it will help me to know from which value of $N_0$, it holds that for every $N > N_0$: $|N!/e - D_N| < 2^{-k}$ for some $k$.

Any help will be appreciated!