One-Ninth Constant

Author

Eric W. Weisstein
July 2, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/One-NinthConstant.nb.

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Value

c=k/.FindRoot[EllipticK[k^2]==2EllipticE[k^2],{k,.9},WorkingPrecision->110]

Λ=Exp[-((π EllipticK[1-c^2])/(EllipticK[c^2]))]

Varga's Constant

1/Λ

V7

Λ=First[Exp[-((π EllipticK[1-c^2])/(EllipticK[c^2]))]/.FindInstance[EllipticK[c^2]==2EllipticE[c^2],c]//FullSimplify]

N[Λ,10^3]

Expressions

(∑_(k=0)^∞)(((2 k+1))^2) (((-x))^((1/2) k (k+1)))

V6

Sum[(((2k+1))^2)(((-x))^(k(k+1)/2)),{k,0,∞}]

V7

TimeConstrained[Sum[(((2k+1))^2)(((-x))^(k(k+1)/2)),{k,0,∞}],60]//Timing

x/.FindRoot[Sum[(((2k+1))^2)(((-x))^(k(k+1)/2)),{k,0,∞}]==0,{x,.1},WorkingPrecision->20]//Timing

(∑_k^∞)-((((-x))^k)/((((((-x))^k)+1))^2))

V6

Sum[-((((-x))^k)/(((1+(((-x))^k)))^2)),{k,∞}]

V7

Sum[-((((-x))^k)/(((1+(((-x))^k)))^2)),{k,∞}]

Series[Sum[-((((-x))^k)/(((1+(((-x))^k)))^2)),{k,100}],{x,0,10}]

Sum[((k (x^k))/(1-(((-x))^k))),{k,∞}]

Series[Sum[((k (x^k))/(1-(((-x))^k))),{k,100}],{x,0,10}]

Series[(1/8)(1-((((2/π)))^2)(2EllipticE[k^2]-EllipticK[k^2])EllipticK[k^2])/.k->(((EllipticTheta[2,0,q])^2)/((EllipticTheta[3,0,q])^2)),{q,0,10}]

Total/@Table[(((-1))^n)#(((-1))^(#))&/@Divisors[n],{n,10}]