1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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##

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

    RUber

    User Avatar
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted