Let $n \in \Bbb N$ be an integer. We want to compute $$\sum_{k=0}^\infty \frac{1}{(nk)!} $$
This clearly makes me think of the Taylor's series of $\exp$ and I know the answer is linked to $e$ but I think I would need to have $\sum_{k=0}^\infty \frac{1}{((n-j)k)!} $ for all others $j$ which makes the problem a lot harder.
I also thought of using Stirling's approximation above some rank but I would lose a lot of precision in the first terms.
The only remaining tool I can think of is $n!=\Gamma(n+1)$ but don't know how useful it can be.
