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Nested Radical

A Radical of the form

\begin{displaymath}
\sqrt{n+\sqrt{n+\sqrt{n+\ldots}}}.
\end{displaymath} (1)

For this to equal a given Real Number $x$, it must be true that
\begin{displaymath}
x=\sqrt{n+\sqrt{n+\sqrt{n+\ldots}}}=\sqrt{n+x},
\end{displaymath} (2)

so
\begin{displaymath}
x^2=n+x
\end{displaymath} (3)

and
\begin{displaymath}
n=x(x-1).
\end{displaymath} (4)

Nested radicals appear in the computation of Pi,
\begin{displaymath}
{2\over\pi} = \sqrt{{\textstyle{1\over 2}}}\sqrt{{\textstyle...
...}}+{\textstyle{1\over 2}}\sqrt{{\textstyle{1\over 2}}}}}\cdots
\end{displaymath} (5)

and in Trigonometrical values of Cosine and Sine for arguments of the form $\pi/2^n$, e.g.,
$\displaystyle \sin\left({\pi\over 8}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sqrt{2-\sqrt{2}}$ (6)
$\displaystyle \cos\left({\pi\over 8}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sqrt{2+\sqrt{2}}$ (7)
$\displaystyle \sin\left({\pi\over 16}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sqrt{2-\sqrt{2+\sqrt{2}}}$ (8)
$\displaystyle \cos\left({\pi\over 16}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sqrt{2+\sqrt{2+\sqrt{2}}}.$ (9)

See also Continued Square Root, Square Root


References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 14-20, 1994.




© 1996-9 Eric W. Weisstein
1999-05-25