Nested Radical Constant

Author

Eric W. Weisstein
September 7, 2008

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

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

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

Analytic

rad=Table[Fold[Sqrt[#1+#2]&,0,Range[n,1,-1]],{n,5}]

<<Utilities`Typesetting`

rad//RootReduce//Column//TraditionalForm

polys=RootReduce/@%//Column

CoefficientList[First[#][x],x^2]&/@Rest[polys]//Column

Flatten[%]

Flatten[Reverse/@(CoefficientList[First[#][x],x^2]&/@Rest[polys])]

Numeric Value

N[Fold[Sqrt[#1+#2]&,0,Range[200,1,-1]],20]

N[Fold[Sqrt[#1+#2]&,0,Range[1200,1,-1]],1000]

Plot

MakeBoxes[label,TraditionalForm]:=SqrtBox[RowBox[{"1","+",SqrtBox[RowBox[{"2","+",SqrtBox[RowBox[{"3","+",SqrtBox[RowBox[{"…","+",SqrtBox["n"]}]]}]]}]]}]]

<<Utilities`Plot`

IntegerListPlot[Table[Fold[Sqrt[#1+#2]&,0,Range[n,1,-1]],{n,15}],PlotRange->All,AxesOrigin->{0,0},
AxesLabel->TraditionalForm/@{n,label},Prolog->{{Blue,Dashing[{.02}],Line[{{0,#},{20,#}}]&[1.7579]}}]