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Proving convergence of integral

  1. Dec 3, 2014 #1
    1. The problem statement, all variables and given/known data
    Prove the following double integral is convergent.

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

    3. 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##

    ##\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.
  2. jcsd
  3. Dec 3, 2014 #2


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    Homework Helper

    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.
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