David Wilson>If you considered
f(n, x) = SUM(n = 0..inf, 1 / (x^n + 1))
I think you mean f(1,x).
to be a closed form, could we find a closed form for
f(a, b) = SUM(n = 0..inf, 1 / (a^n + b^n))?
Mathematica doesn't think so, even with its all-purpose In[86]:= Sum[1/(1 - a*x^n), n] Out[86]= -((-n Log[x] + QPolyGamma[0, n - Log[1/a]/Log[x], x])/Log[x]) (Note indefinite sum.) Which can even, with some work, express the ever-nagging Out[139]= (1/(4 ArcCsch[2])) Sqrt[5] (-QPolyGamma[0, 1, GoldenRatio^2] + I (QPolyGamma[0, 1/2 - (I π)/(4 ArcCsch[2]), 1/GoldenRatio^2] - QPolyGamma[0, 1/2 + (I π)/(4 ArcCsch[2]), 1/GoldenRatio^2]) + Re[QPolyGamma[0, 1 - (I π)/(2 ArcCsch[2]), GoldenRatio^2]]) In[140]:= N[{%, Sum[1/Fibonacci[n], {n, ∞}]}] Out[140]= {3.35989 + 0. I, 3.35989} --rwg