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.


  • 1
    $\begingroup$ I highly doubt either $\sum_p1/p$ or $\sum_n1/(n^2)!$ has a sensible closed form. $\endgroup$ – Wojowu Jan 25 at 16:53
  • $\begingroup$ @JackD'Aurizio Forgot an exclamation mark! $\endgroup$ – Wojowu Jan 25 at 17:21

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