I'm trying to evaluate the equation below excluding the case when $n_x=n_y=n_z=0$. I know this equation converges everywhere except where x,y, and z are all multiples of $2\pi$. I've attempted breaking the summation into 3 parts which avoid the unwanted case. Unfortunately Mathematica wasn't able to solve that. I also tried defining a function to use in the place of the fraction with a value of 0 when $n_x=n_y=n_z=0$. That hasn't worked either. I've also been trying to break the problem down into smaller pieces and looking at special cases with little success. Any help at all would be appreciated. $$\sum_{n_x=0}^\infty \sum_{n_y=0}^\infty \sum_{n_z=0}^\infty \frac{\cos(n_x x)\cos(n_y y)\cos(n_z z)}{n_x^2+n_y^2+n_z^2}$$

This equation would give what I call the Madelung Potential. Essentially the potential produced by a positive point charge in a cubic lattice submerged in a sea of negative charge so that the total charge is 0.


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