Foias Constant

Author

Eric W. Weisstein
April 16, 2004

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

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Foias Problem (with typo ;)

Attempt to Solve Exactly

RSolve[x[n+1]==(1+1/x[n])^n,x[n],n]

x[1]:=a
x[n_]:=x[n]=(1+1/x[n-1])^(n-1)

α=SetPrecision[1.187452351126501,16]

Plot[Evaluate[x/@Range[20]],{a,0,3},
PlotStyle->{Red,Blue},
AxesLabel->TraditionalForm/@{HoldForm[α],(Subscript[x, n])[HoldForm[α]]},
PlotRange->{0,10},
Epilog->{Green,Line[{{α,0},{α,1}}]}]//Timing

Plot[x[40,a]-x[40,a+.000001],{a,1.18744,1.18746}]

f=D[1/x[50,a],a];

Plot[f,{a,1.15,1.2},PlotRange->All]

FindRoot[f==10^(-57),{a,1.186,1.19},AccuracyGoal->Infinity,WorkingPrecision->70,MaxIterations->20]

Original Problem

Attempt to Solve Recurrence Equation

RSolve[{y[1]==a,y[n]==(1+1/y[n-1])^y[n-1]},y[n],n]

Plot of Solutions

y[1]:=a
y[n_]:=y[n]=(1+1/y[n-1])^y[n-1]

Plot[Evaluate[y/@Range[5]],{a,0,3},
PlotStyle->{Red,Yellow,Green,Blue,Violet},
AxesLabel->TraditionalForm/@{β,(Subscript[x, n])[β]}]//Timing

Attempt to find analytic solution for (Subscript[x, ∞])

V6

Solve[x==(1+1/x)^x,x]

Solve[Log[x]==x Log[1+1/x],x]

Reduce[x==(1+1/x)^x,x,#]&/@{Integers,Reals,Complexes}

V7

FindInstance[x==(1+1/x)^x,x,Reals]

N[Root[{-(((1+(1/(#1))))^(#1))+#1&,2.2931662874118610315080282912508058643722572903271212485377`30.}],1000]

Numeric Solution

β=x/.FindRoot[x==(((1+(1/x)))^x),{x,2},WorkingPrecision->110]