Nested Radical

A Radical of the form

$$\sqrt{n+\sqrt{n+\sqrt{n+\ldots}}}.$$ (1)

For this to equal a given Real Number $x$, it must be true that

$$x=\sqrt{n+\sqrt{n+\sqrt{n+\ldots}}}=\sqrt{n+x},$$ (2)

so

$$x^2=n+x$$ (3)

and

$$n=x(x-1).$$ (4)

Nested radicals appear in the computation of Pi,

$${2\over\pi} = \sqrt{{\textstyle{1\over 2}}}\sqrt{{\textstyle... ...}}+{\textstyle{1\over 2}}\sqrt{{\textstyle{1\over 2}}}}}\cdots$$ (5)

and in Trigonometrical values of Cosine and Sine for arguments of the form $\pi/2^n$, e.g.,

$$\sin\left({\pi\over 8}\right) = {\textstyle{1\over 2}}\sqrt{2-\sqrt{2}}$$ (6)

$$\cos\left({\pi\over 8}\right) = {\textstyle{1\over 2}}\sqrt{2+\sqrt{2}}$$ (7)

$$\sin\left({\pi\over 16}\right) = {\textstyle{1\over 2}}\sqrt{2-\sqrt{2+\sqrt{2}}}$$ (8)

$$\cos\left({\pi\over 16}\right) = {\textstyle{1\over 2}}\sqrt{2+\sqrt{2+\sqrt{2}}}.$$ (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.