Liouville's Constant

Author

Eric W. Weisstein
April 11, 2006

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

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Sequence

Length[l=RealDigits[Sum[2^(-n!),{n,8}],2][[1]]]

ArrayPlot[Table[l[[x]]-l[[y]],{x,256},{y,256}],PixelConstrained->True]

Value

V6

Sum[10^(-n!),{n,∞}]

V7

Sum[10^(-n!),{n,∞}]

SumConvergence[10^(-n!),n]

Numerical

N[Sum[10^(-n!),{n,10}],1000]

N[Sum[10^(-n!),{n,11}],1000]

NSum[10^(-n!),{n,∞}]//Timing

NSum[10^(-n!),{n,∞},WorkingPrecision->5]//Timing

NSum[10^(-n!),{n,∞},WorkingPrecision->20]//Timing

NSum[10^(-n!),{n,∞},WorkingPrecision->30]//Timing

Continued Fraction

Take[c=ContinuedFraction[Sum[10^(-n!),{n,9}],256],50]//Timing

LogListPlot[Length[IntegerDigits[#]]&/@Rest[c],PlotRange->All,PlotJoined->True,PlotStyle->Red,AxesLabel->TraditionalForm/@{n,d[(Subscript[a, n])]}]

HighWaterMarks[Rest[c]]//Last

Length/@IntegerDigits/@First[HighWaterMarks[Rest[c]]]

Table[(n-1)n!,{n,8}]

Approximate Identity

approx=N[x/.{ToRules[Roots[10x^6-75x^3-190x+21==0,x]]},20]

approx[[1]]-L

N[10x^6-75x^3-190x+21/.x->L]

N[Root[10x^6-75x^3-190x+21,1],100]

RealDigits[%]