Benson's Formula

Author

Eric W. Weisstein
April 16, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/BensonsFormula.nb.

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

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

Constant

With[{n=25},
Plus@@Flatten[Table[If[Equal[0,i,j,k],0,((((-1.))^(i+j+k))/(Sqrt[i^2+j^2+k^2]))],{i,-n,n},{j,-n,n},{k,-n,n}]]
]//Timing

-12Pi NSum[Sech[Sqrt[m^2+n^2]Pi/2]^2,{m,1,∞,2},{n,1,∞,2},
WorkingPrecision->50]//Timing

-12Pi NSum[Sech[Sqrt[m^2+n^2]Pi/2]^2,{m,1,∞,2},{n,1,∞,2},
NSumTerms->200,WorkingPrecision->105]//Timing