False Logarithmic Series

Author

Eric W. Weisstein
September 22, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/FalseLogarithmicSeries.nb.

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

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

Additional Authors

FalseLogarithm by Oleksander Pavlyk

Definition

s(a,x)==(∑_n^∞)(((∏_(k=0)^(n-1))(1-x (a^(-k))))/(1-(a^n)))

s[a_,x_]:=Sum[(1/(1-(a^n)))Product[1-x (a^(-k)),{k,0,n-1}],{n,∞}]

s[a,x]

s(a,0)==((QPolyGamma[1, (1/a)])/(ln(a)))+(Subscript[log, a])(a-1)-1

s[a,0]

Sum[1/(1 - a^n), {n, Infinity}]

FullSimplify[((QPolyGamma[1,1/a])/(Log[a]))+Log[a,a-1]-1,a>1]//TraditionalForm

FalseLogarithm[a_?NumericQ,x_?NumericQ]:=Module[{sum=0,apow=a,term=1-x,oldsum},
Catch[While[True,oldsum=sum;
sum+=Divide[term,(1-apow)];
If[sum==oldsum,Throw[sum]];
term*=(1-Divide[x,apow]);
apow*=a;]]
]/;(Precision[{a,x}]<Infinity&&Abs[a]>1)||IntegerQ[Log[a,x]]

With[{a=2},Table[{s[a,a^n],FalseLogarithm[a,a^n],Log[a,a^n]},{n,4}]]

Plot

With[{a=2},Plot[{FalseLogarithm[a,(a^x)],Log[a,(a^x)]},{x,-2.1,5},PlotRange->All,PlotStyle->{Red,Blue},
Epilog->{PointSize[.015],Point[Table[{k,k},{k,-2,5}]]}]]