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Definite Triple Integral to Series

  1. Mar 24, 2013 #1
    1. The problem statement, all variables and given/known data

    Does the triple integral
    [itex]\int^{1}_{0}\int^{1}_{0}\int^{1}_{0}\frac{1}{1+x^2 y^2 z^2}[/itex] = [itex]\sum^{∞}_{n=0}\frac{1}{(2n+1)^3}[/itex]

    2. Relevant equations



    3. The attempt at a solution
    I've not a single clue on what to do with this problem. I figured maybe I could find a decent conversion of variables and find the Jacobian and switch variables to make this easier but tht didn't seem to work.

    I don't know why but I think the series expansion of arcsin or arctan is needed here, but I'm not exactly sure how to incorporate it in. I don't even know how to evaluate this triple integral..
     
  2. jcsd
  3. Mar 24, 2013 #2

    jbunniii

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    Hint: the integrand is ##\leq 1##, so does that give you an upper bound for the left hand side? Notice also that the first term of the series on the right hand side is ##1##, so that gives you a lower bound for the right hand side.
     
    Last edited: Mar 24, 2013
  4. Mar 24, 2013 #3
    Ohh I think I get what you mean

    The upper bound for the LHS is 1, and the first term of the right hand side is 1 and then adding constants, so they can't be equal?

    There's only one intersection of their ranges so it doesnt work out then I guess?
     
  5. Mar 24, 2013 #4

    jbunniii

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    That's right, the LHS is ##\leq 1##, and the RHS is ##> 1##, so they can't be equal.
     
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