Definite Triple Integral to Series

[Elysian]

Homework Statement

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

Homework Equations

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..

 

[jbunniii]

Homework Statement

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

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.

 

[Elysian]

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.

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 doesn't work out then I guess?

 

[jbunniii]

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?

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