Ramanujan ψ Sum

Author

Eric W. Weisstein
November 12, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/RamanujanPsiSum.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RamanujanPsiSum.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Sum

ψ[a_]:=1+2Sum[1/((a k)^3-a k),{k,Infinity}];
(Subscript[ψ, 2])[a_]:=-(PolyGamma[0,1/a]+PolyGamma[0,1-1/a]+2 EulerGamma)/a;

ψ[a]

FullSimplify[ψ[a]]

Table[ψ[n],{n,2,8}]

FullSimplify[Table[ψ[n],{n,2,8}]]

FullSimplify[Expand[Table[(Subscript[ψ, 2])[n],{n,2,8}]]]

{ψ[5],(Subscript[ψ, 2])[5],(Log[GoldenRatio]/Sqrt[5]+Log[5]/2)}//N

{Log[432]/6,Log[3]/2+2Log[2]/3}//N