Convergence of a particular double series For the double series
$$ \sum_{m,n=1}^{\infty} \frac{1}{(m+n)^p} , $$
I was wondering when it converges. I want to use double integrals to estimate it, but I don't know how to write the process accurately... Could you show me a detailed computation? Thanks!

While we wait for someone to do it using double integrals, here's another way:
Given a positive integer $k$, the number of pairs $m,n$ with $m\ge1$, $n\ge1$, and $m+n=k$ is $k-1$. So your sum is $\sum_{k=2}^{\infty}(k-1)/k^p$, and the usual single-series methods apply. 
