Khinchin Harmonic Mean

Author

Eric W. Weisstein
December 10, 2017

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

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

©2017 Wolfram Research, Inc. except for portions noted otherwise

(Subscript[K, -1])

Entity["MathematicalConstant","KhinchinHarmonicMeanConstant"]["DefiningFormula"]

Entity["MathematicalConstant","KhinchinHarmonicMeanConstant"]["SeriesRepresentations"]

Entity["MathematicalConstant","Khinchin"]["DefiningFormula"]

Entity["MathematicalConstant","Khinchin"]["SeriesRepresentations"]

((log(2))/((∑_n^∞)((log(1-(1/(((n+1))^2))))^(-1/n))))

Sum[Log[(1-1/(n+1)^2)^(-1/n)],{n,∞}]

Sum[-((Log[(1-1/(n+1)^2)])/n),{n,∞}]

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

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

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

plouffe=1.745405662407346863494596309683661067294936618777984;

A087491

sloane=1.74540566240734686349459630968366106729493661877798425659501377351607857522087342565205788645678324242`100.;

sloane==plouffe

sloane

NSumTerms

terms3=Log[2]/NSum[Log[(1-1/(n+1)^2)^(-1/n)],{n,∞},NSumTerms->10^3,WorkingPrecision->500,PrecisionGoal->110]

terms3-sloane

(terms4=Log[2]/NSum[Log[(1-1/(n+1)^2)^(-1/n)],{n,∞},NSumTerms->10^4,WorkingPrecision->250,PrecisionGoal->110])//Timing

terms4-sloane

(terms42=Log[2]/NSum[-Log[(1-1/(n+1)^2)]/n,{n,∞},NSumTerms->10^4,WorkingPrecision->250,PrecisionGoal->110])//Timing

terms4-terms42

RealDigits[Log[2]/NSum[Log[(1-1/(n+1)^2)^(-1/n)],{n,∞},NSumTerms->10^4,WorkingPrecision->250,PrecisionGoal->110]][[1,;;100]]

terms5=Log[2]/NSum[-Log[(1-1/(n+1)^2)]/n,{n,∞},NSumTerms->10^5,WorkingPrecision->500,PrecisionGoal->200]

terms5-sloane

terms4-terms5

terms6=Log[2]/NSum[-Log[(1-1/(n+1)^2)]/n,{n,∞},NSumTerms->10^6,WorkingPrecision->500,PrecisionGoal->200]

terms6-sloane

terms5-terms6

Explicit numeric summation

Log[2]/Sum[Log[(N[1,100]-1/(n+1)^2)^(-1/n)],{n,(10^3)}]//Timing

Log[2]/Sum[Log[(N[1,100]-1/(n+1)^2)^(-1/n)],{n,(10^4)}]//Timing

Log[2]/Sum[Log[(N[1,100]-1/(n+1)^2)^(-1/n)],{n,(10^5)}]//Timing

(k5=((Log[2])/(Sum[-((Log[(N[1,100]-1/(n+1)^2)])/n),{n,(10^5)}])))//Timing

(k6=((Log[2])/(Sum[-((Log[(N[1,100]-1/(n+1)^2)])/n),{n,(10^6)}])))//Timing

k6-k5

WynnEpsilon

NSum[Log[(1-1/(n+1)^2)^(-1/n)],{n,∞},Method->"WynnEpsilon",NSumTerms->10^5,WorkingPrecision->500,PrecisionGoal->200]

Log[2]/%

NSum[-Log[(1-1/(n+1)^2)]/n,{n,∞},Method->"WynnEpsilon",NSumTerms->10^5,WorkingPrecision->500,PrecisionGoal->200]

EulerMaclaurin

NSum[Log[(1-1/(n+1)^2)^(-1/n)],{n,∞},Method->"EulerMaclaurin",WorkingPrecision->500,PrecisionGoal->100]

Log[2]/%

NSum[Log[(1-1/(n+1)^2)^(-1/n)],{n,∞},Method->"EulerMaclaurin",WorkingPrecision->500,PrecisionGoal->200]

Precision[%103]

Log[2]/%

Default

sum1=NSum[Log[(1-1/(n+1)^2)^(-1/n)],{n,1,1000},WorkingPrecision->500,PrecisionGoal->100]//N[#,50]&

sum2=NSum[Log[(1-1/(n+1)^2)^(-1/n)],{n,1000+1,∞},WorkingPrecision->500,PrecisionGoal->100]

N[Log[2]/(sum1+sum2),50]

RealDigits[Log[2]/NSum[Log[(1-1/(n+1)^2)^(-1/n)],{n,∞},WorkingPrecision->500,PrecisionGoal->100],10,50][[1]]

{1,7,4,5,4,0,5,6,6,2,4,0,7,3,4,6,8,6,3,4,9,4,5,9,6,3,0,9,6,8,3,6,6,1,0,6,7,2,9,4,9,3,6,6,1,8,7,7,7,9,8,4,2,5,6,5,9,5,0,1,3,7,7,3,5,1,6,0,7,8,5,7,5,2,2,0,8,7,3,4,2,5,6,5,2,0,5,7,8,8,6,4,5,6,7,8,3,2,4,2,4,2}

sum=Sum[Log[(1-1/(n+1)^2)^(-1/n)],{n,10^5}];//Timing

N[Log[2]/sum,50]

plouffe