For natural number $n$, we set $r_n = \sum_{k=n+1}^{+\infty} \frac{1}{k!}$. Prove the bound $$r_n \leq \frac{1}{(n+1)!} \sum_{k=0}^{+\infty} \frac{1}{(n+2)^k}$$ Deduce a simple equivalent of $r_n$ as $n$ tends to $+\infty$.
For natural number $n$, we set $r_n = \sum_{k=n+1}^{+\infty} \frac{1}{k!}$. Prove the bound
$$r_n \leq \frac{1}{(n+1)!} \sum_{k=0}^{+\infty} \frac{1}{(n+2)^k}$$
Deduce a simple equivalent of $r_n$ as $n$ tends to $+\infty$.