I am trying to analyze the convergence of multidimensional infinite sums such as those in the following form:
$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{1}{1+\alpha\exp (\beta n) \exp (\gamma m)}$
where $\alpha,\beta,\gamma\in(0,\infty)$.
I'm quickly realizing though, that I have no experience with analyzing sums like this. I realize that because $\mathbb{N}^2$ is countable, I could in principle re-write this sum as a related one:
$\sum_{n=0}^{\infty} f(n)$
where $f(n)$ could even be guarenteed monotone decreasing. However, I've been having trouble getting anywhere with this.
Is there some well-established method of attacking convergence of multidimensional sums like this one? If so, I would very much appreciate being pointed in the right direction.