Mathematica MATHEMATICA, with a triple sum

[ggold1]

I'm having trouble solving this summation and was wondering if anyone could put me on the right track. Here's the problem below, I've also attached a pdf of it.

Sum from 1 to infinity in steps of 2 for l, m, and n:((-1)^((m - 1)/2))*(m/(n^2 + l^2 + m^2))

I've tried using the built in sum functions

Sum[((-1)^((m - 1)/2))*(m/(n^2 + l^2 + m^2)),{l,1,Infinity,2},{m,1,Infinity,2},{n,1,Infinity,2}]

but mathematica continuously sums this and never reaches an approximation. Is there any function I'm missing or a way to simplify this?

 

[Bill Simpson]

Sometimes something like this will give you a result
Limit[Sum[((-1)^((m-1)/2))*(m/(n^2+l^2+m^2)), {l,1,i, 2}, {m,1,i,2}, {n,1,i,2}], i -> Infinity]
but you need to carefully consider whether that is a valid change to your problem.