Hi! I need to show that this double series diverge:

I first used the Integral Test on the inner summation (letting m fixed) with the improper integral

) and concluded it was finite, then computed a second improper integral with the integrand being my previous result (i.e

) and found this was diverging, and hence that the double series diverges as well.

But I think i get the right answer by accident: first I have never seen anywhere a "double improper integral test" and, more seriously, i know that the integral test tells us that if the integral converges then so does the series, but it doesn't tell that the series converges to the same value as the integral, yet i am using this value in my second integral...

I also need to show that

is finite and used the same method as above to find that it was indeed finite.

So is this approach justified? If not, how can we determine convergence and divergence for double summations?