Proving convergence of integral

[Panphobia]

Homework Statement

Prove the following double integral is convergent.

$\int_0^1 \int_0^1 \frac{1}{1-xy}\, dx \, dy$

The Attempt at a Solution

This was a bonus question on my final exam in calc 3 yesterday, I just want to show my steps and see if they were right.

So I realized that
$\frac{1}{1 - xy} = \sum_{n=1}^\infty x^ny^n$

So
$\int_0^1 \int_0^1 \sum_{n=1}^\infty x^ny^n\, dx \, dy = \sum_{n=1}^\infty\frac{1}{(n+1)^2}$
Then I used the direct comparison test to show that

$\frac{1}{(n+1)^2} \lt \frac{1}{n^2}$
So since it is smaller than a convergent p-series, it is also convergent.

 

[RUber]

Thanks for sharing this problem.
My only note would be that (I think) a series needs to be convergent in order to pass it through the integral. I would first take the partial sum to some large (finite) M, then you can pass it through and take the limit of the partial sums as M goes to infinity.

Aside from that, your work looks good. Nice and clean.