Is there a comprehensive list of convergence tests for double series?

Question

I’m looking for a comprehensive list of methods to check for convergence of infinite double series. Specifically conditional convergence.

Answer 1

First of all, you have to be careful with exactly what you mean by "infinite double series", because rearrangements of conditionally convergent series can diverge. I'll assume that when you talk about convergence of a double series of the form

$$ \sum_{i=1}^\infty \sum_{j=1}^\infty a_{ij}$$ you mean that for each $i$, the series $\sum_{j=1}^\infty a_{ij}$ converges to a value $b_i$, and the series $\sum_{i=1}^\infty b_i$ converges.

The methods for proving convergence of such double series are essentially the same as those for a single series, except that it may be more difficult because you might not know $b_i$ in closed form.