Sum[1/Product[1+1/k^2 ,{k, n}]/n^2,{n,Infinity}] == 1 -1/Product[1+1/k^2,{k,Infinity}] ==1-Pi/Sinh[Pi] and, playing in the same garden, I suppose the following is generally known (among those that know such things) Sum[HarmonicNumber[n]/n^w,{n,\[Infinity]}] == Sum[-Zeta[k]Zeta[1+w-k],{k,2,w-1}]/2+(w+2)/2 Zeta[w+1] but I can't find it in http://mathworld.wolfram.com/RiemannZetaFunction.html nor in http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/ With thanks to Olivier Gerard and Bill Gosper for their patient tutoring. Wouter.