- #1
Panphobia
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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.