Nested Sigma Noation

I hope you've seen these before; they're the basic definitions of integrals. In the second one, I have nested an integral within an integral, thus producing what's known as a double integral. The alternate integral notation reveals the answer to two of your three questions:

"How do you find the sum of this?"
"What happens when you add a third? A fourth...?"

Just like in a double-integral, work your way outward from the innermost integral. The exact same process occurs here with your nested sums:





The convergence/divergence tests are essentially the same as those for single summations. If one of the series produced diverges, the nested integral will diverge. For example, since your series includes a p-series of p=1, the series diverges. Alternatively, you can switch the order:





You can easily see that the series of 1's sums up to infinity so the series diverges. Unfortunately, I haven't really worked with summations so I can't tell you if there exists series that might diverge despite being composed of individually convergent series. I'm fairly certain that there exist such series, but my knowledge of this subject ends about here. I'm sure someone more qualified in this field such as Noah or Russt can further help you out.