In[679]:= Sum[(-1)^n/n^2*Cos[Sqrt[\[Pi]^2*n^2 - 9]], {n, \[Infinity]}] Out[679]= -(\[Pi]^2/(12 E^3)) How the hockeysticks did it know that? It's one of the weirder formulas in www.tweedledum.com/rwg/idents.htm, except with the summand written more sanely, exactly as in (1.2) of https://projecteuclid.org/download/pdf_1/euclid.ijm/1255987146 But (1.2) is the b = 3i case of (1.1): In[680]:= FunctionExpand@Sum[(-1)^n/n^2*Cos[Sqrt[\[Pi]^2*n^2 + b^2]], {n, \[Infinity]}] == \[Pi]^2/4*(Sin[b]/b - Cos[b]/3) Out[680]= n 2 2 2 (-1) Cos[Sqrt[b + n Pi ]] 1 2 Cos[b] Sin[b] Sum[----------------------------, {n, Infinity}] == - Pi (-(------) + ------) 2 4 3 b n which, as you can see, stumps FunctionExpand. Why would anyone code up this very special case of an obscure identity? To snow a casual observer? To get a rise out of yours truly?? WRI works in mysterious ways. But they stopped short. If only they had (1.1), they could've expanded at b=3i into an infinitude of obscurities: {Sum[((-1)^n*Sin[Sqrt[-9 + n^2*Pi^2]])/(n^2*Sqrt[-9 + n^2*Pi^2]), {n, Infinity}] == -(((-7 + E^6)*Pi^2)/(216*E^3)), Sum[(9*(-1)^n*Cos[Sqrt[-9 + n^2*Pi^2]])/(2* n^2*(-9 + n^2*Pi^2)), {n, Infinity}] + Sum[-(((-1)^n*Pi^2* Sin[Sqrt[-9 + n^2*Pi^2]])/(2*(-9 + n^2*Pi^2)^(3/2))), {n, Infinity}] == (55*Pi^4*Sinh[6])/31104, Sum[((-1)^n*Pi^2*Cos[Sqrt[-9 + n^2*Pi^2]])/(-9 + n^2*Pi^2)^2, {n, Infinity}] + Sum[((-1)^n*(-27 + 2*n^2*Pi^2)* Sin[Sqrt[-9 + n^2*Pi^2]])/(n^2*(-9 + n^2*Pi^2)^(5/2)), {n, Infinity}] == (5*(-7 + E^6)*Pi^2)/(1944*E^3)} --rwg