Evaluating a double sigma

Question

Evaluate $$\sum_{m=0}^{\infty} \sum_{n=0}^{\infty}\frac{m!n!}{(m+n+2)!}$$ How do I start with the problem? Infinite sum of factorials?

Answer 1

Evaluating a double sigma. How do I start with the problem ?

By evaluating a simple sigma! ;-$)$ Could you prove that the inner sum evaluates to $\dfrac1{(m+1)^2}$ ?

My bet is that it telescopes, due to some clever algebraic manipulations of binomial coefficients.

Afterwards, see Basel problem and the Riemann $\zeta$ function.