Erdős-Borwein Constant

Author

Eric W. Weisstein
August 1, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/Erdos-BorweinConstant.nb.

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Analytic

(∑_n^∞)(1/((2^n)-1))==1-((QPolyGamma3[0, 1, (1/2)])/(log(2)))

NSum[(1/((2^n)-1)),{n,∞},WorkingPrecision->20]

V6

Sum[(1/((2^n)-1)),{n,∞}]

Sum[(1/(2^(m n))),{m,∞},{n,∞}]

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

V7

Sum[(1/((2^n)-1)),{n,∞}]

FullSimplify[%]

N[%]

(∑_m^∞)(∑_n^∞)(1/(2^(m n)))==1-((QPolyGamma3[0, 1, (1/2)])/(log(2)))

NSum[(1/(2^(m n))),{m,∞},{n,∞},WorkingPrecision->10]

V7

Sum[(1/(2^(m n))),{m,∞},{n,∞}]//Timing

(∑_m^∞)(((2^m)+1)/((2^(m^2)) ((2^m)-1)))

NSum[(1/(2^(m^2)))(((2^m)+1)/((2^m)-1)),{m,∞}]

V7

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

NSum[(1/(2^(m^2)))(((2^m)+1)/((2^m)-1)),{m,∞}]

1-((QPolyGamma[0,1,(1/2)])/(Log[2]))//N

V8

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

(∑_n^∞)((DivisorSigma[0, n])/(2^n))

NSum[((DivisorSigma[0,n])/(2^n)),{n,∞},AccuracyGoal->20,PrecisionGoal->Infinity,WorkingPrecision->200]

V7

Sum[((DivisorSigma[0,n])/(2^n)),{n,∞}]

1-((QPolyGamma[0,1,(1/2)])/(Log[2]))//N

V8

Sum[((DivisorSigma[0,n])/(2^n)),{n,∞}]

Series Transformation

Sum[((x^n)/(1-(x^n))),{n,∞}]

This is a Lamber series

Series[Sum[((x^n)/(1-(x^n))),{n,200}],{x,0,20}]

Sum[DivisorSigma[1,n]x^n,{n,∞}]

CoefficientList[%,x]

Table[If[n==0,0,DivisorSigma[0,n]],{n,0,20}]

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

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

Generalization

(∑_(q=1)^∞)(1/((q^n)-r))

V7

Sum[(1/((q^n)-r)),{q,1,∞}]//Timing