It's possible, that equation $\sum\_{n} n!^s=1+2\sum\_n (2n+1)!^s$ is correct for all $s \in \mathbb{R}$ with which sum $\sum\_{n} n!^s$ is convergent?


I'm looking for closed formula of that sum and correctness of that equation is very important to me. Thanks for help.



Correctness of equation for $\sum_{n} n!^s$
Assuming that your sums run over the nonnegative integers, 


$$\sum\_{n \ge 0} n!^s$$


is convergent if and only if $s < 0$. (Apply the ratio test.)


It's easy to evaluate both sides at $s = -1$. In that case, the left-hand side is just 


$$ \sum\_{n \ge 0} {1 \over n!} = e$$


and the left-hand side is $1 + 2 \sinh 1$, where $\sinh$ is the hyperbolic sine. In particular, the sum $\sum\_{n \ge 0} 1/(2n+1)!$ is greater than 1 (the $n=1$ term is 1), and so $1 + 2\sum\_{n \ge 0} 1/(2n+1)! > 3$. So the identity you want to prove is false.


