# Evaluation of $\sum_{p \text{ is a prime }}\frac{1}{p!}$

I apologize in advance, if these have been posted earlier (as I couldn't find them).

While surfing through the Internet, I came across the following problem:

Evaluate: $$\displaystyle\sum_{n=0}^{\infty} \frac{1}{(3n)!}$$, which I computed successfully by using roots of unity filter.

After a little while I became a bit curious and wondered how to compute the following series: $$\displaystyle\sum_{p \text{ is a prime }}\frac{1}{p!}$$.

My first idea was to use sieve of erasthosthenes along with roots of unity filter but I can't figure out how to do so.

Questions similar to these are also coming to my mind, which I have no idea on how to approach to, like $$\sum_{n=0}^{\infty}\frac{1}{(n^2)!}$$, etc.

If possible, please provide me little hints on how to proceed.

Thanks.

• I highly doubt either $\sum_p1/p$ or $\sum_n1/(n^2)!$ has a sensible closed form. – Wojowu Jan 25 at 16:53
• @JackD'Aurizio Forgot an exclamation mark! – Wojowu Jan 25 at 17:21