Nested Radical

Author

Eric W. Weisstein
February 16, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/NestedRadical.nb.

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

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

Vieta

Rest[NestList[Sqrt[1/2+#/2]&,0,6]]

Times@@%

N[Times@@%-2/Pi,20]

Trigonometric powers of two

FunctionExpand/@Table[{n,Sin[Pi/2^n],Cos[Pi/2^n]},{n,5}]//TableForm

Nested x (Sqrt[x (Sqrt[x])])

r=Nest[x Sqrt[#]&,1,20]/x

Table[#-r/.x->#&[Random[]],{20}]

Nested  (Sqrt[n+ (Sqrt[n])])

r=Sqrt[Nest[n+ Sqrt[#]&,1,15]]

Table[(1+Sqrt[4n+1])/2 -r/.n->#&[Random[]],{20}]

Nested  ((a+ ((a+…)^(1/n)))^(1/n))

r=(Nest[a+#^(1/n)&,1,100])^(1/n);

GoldenRatio

N[r/.{a->1,n->2},10]

Solve[x^2==1+x,x]

N[GoldenRatio,10]

Plastic Constant

N[r/.{a->1,n->3},10]

RootReduce/@Roots[x^3==1+x,x]

N[%[[1,2]],10]

Silver Constant

r=(Nest[7+7#^(1/3)&,1,100])^(1/3);

N[r]

N[2+((r+2)/(r+1)),10]

N[2+2Cos[(2/7)π],10]

Nested (Sqrt[x ((x ((x (x^(1/5)))^(1/4)))^(1/3))])

r=Fold[x #1^(1/#2)&,x,Range[10,2,-1]]/x

Table[x^(E-2) -r/.x->#&[Random[]],{20}]