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Nested Radical

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Nested radicals are expressions of the form

(1)

Herschfeld (1935) proved that a nested radical of real nonnegative terms converges iff is bounded. He also extended this result to arbitrary powers (which include continued square roots and continued fractions as well), a result is known as Herschfeld's convergence theorem.

Nested radicals appear in the computation of pi,

(2)

(Vieta 1593; Wells 1986, p. 50; Beckmann 1989, p. 95), in trigonometrical values of cosine and sine for arguments of the form , e.g.,

			(3)
			(4)
			(5)
			(6)

The iteration in these radicals inverts the restriction of to and sends to for , where is the second-degree Chebyshev polynomial of the first kind (Moreno and García-Caballero 2012).

Nested radicals also appear in the computation of the golden ratio

(7)

and plastic constant

(8)

Both of these are special cases of

(9)

which can be exponentiated to give

$x^n=a+RadicalBox[{a, +, RadicalBox[{a, +, ...}, n]}, n],$

(10)

so solutions are

(11)

In particular, for , this gives

(12)

The silver constant is related to the nested radical expression

(13)

There are a number of general formula for nested radicals (Wong and McGuffin). For example,

$x=RadicalBox[{{(, {1, -, q}, )}, {x, ^, n}, +, q, {x, ^, {(, {n, -, 1}, )}}, RadicalBox[{{(, {1, -, q}, )}, {x, ^, n}, +, q, {x, ^, {(, {n, -, 1}, )}}, RadicalBox[..., n]}, n]}, n]$

(14)

which gives as special cases

(15)

(, , ),

$x=RadicalBox[{{x, ^, {(, {n, -, 1}, )}}, RadicalBox[{{x, ^, {(, {n, -, 1}, )}}, RadicalBox[{{x, ^, {(, {n, -, 1}, )}}, RadicalBox[..., n]}, n]}, n]}, n]$

(16)

(), and

(17)

(). Equation (14) also gives rise to

$q^((n^k-1)/(n-1))x^(n^j)=RadicalBox[{{q, ^, {(, {{(, {{n, ^, {(, {k, +, 1}, )}}, -, n}, )}, /, {(, {n, -, 1}, )}}, )}}, {(, {1, -, q}, )}, {x, ^, {(, {n, ^, {(, {j, +, 1}, )}}, )}}, +, ...}, n] ...+RadicalBox[{{q, ^, {(, {{(, {{n, ^, {(, {k, +, 2}, )}}, -, n}, )}, /, {(, {n, -, 1}, )}}, )}}, {(, {1, -, q}, )}, {x, ^, {(, {n, ^, {(, {j, +, 2}, )}}, )}}, +, RadicalBox[..., n]}, n]^_,$

(18)

which gives the special case for , , , and ,

sqrt(2)=sqrt(2/(2^(2^0))+sqrt(2/(2^(2^1))+sqrt(2/(2^(2^2))+sqrt(2/(2^(2^3))+sqrt(2/(2^(2^4))+...))))).

(19)

Equation (17) can be generalized to

$x^(1/(n-1))=RadicalBox[{x, RadicalBox[{x, RadicalBox[{x, ...}, n]}, n]}, n]$

(20)

for integers , which follows from

			(21)
			(22)
			(23)
			(24)
			(25)

In particular, taking gives

(26)

(J. R. Fielding, pers. comm., Oct. 8, 2002).

Ramanujan discovered

x+n+a=sqrt(ax+(n+a)^2+xsqrt(a(x+n)+(n+a)^2+...))
...+(x+n)sqrt(a(x+2n)+(n+a)^2+(x+2n)sqrt(...))^_^_,

(27)

which gives the special cases

(28)

for , (Ramanujan 1911; Ramanujan 2000, p. 323; Pickover 2002, p. 310), and

(29)

for , , and . The justification of this process in general (and in the particular example of , where is Somos's quadratic recurrence constant) is given by Vijayaraghavan (in Ramanujan 2000, p. 348).

An amusing nested radical follows rewriting the series for e

(30)

(31)

$x^(e-2)=sqrt(xRadicalBox[{x, RadicalBox[{x, RadicalBox[{x, ...}, 5]}, 4]}, 3])$

(32)

(J. R. Fielding, pers. comm., May 15, 2002).

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References

Beckmann, P. A History of Pi, 3rd ed. New York: Dorset Press, 1989.Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 14-20, 1994.Borwein, J. M. and de Barra, G. "Nested Radicals." Amer. Math. Monthly 98, 735-739, 1991.Herschfeld, A. "On Infinite Radicals." Amer. Math. Monthly 42, 419-429, 1935.Jeffrey, D. J. and Rich, A. D. In Computer Algebra Systems (Ed. M. J. Wester). Chichester, England: Wiley, 1999.Landau, S. "A Note on 'Zippel Denesting.' " J. Symb. Comput. 13, 31-45, 1992a.Landau, S. "Simplification of Nested Radicals." SIAM J. Comput. 21, 85-110, 1992b.Landau, S. "How to Tangle with a Nested Radical." Math. Intell. 16, 49-55, 1994.Landau, S. "

: Four Different Views." Math. Intell. 20, 55-60, 1998.Moreno, S. G. and García-Caballero, E. M. "Chebyshev Polynomials and Nested Square Roots." J. Math. Anal. Appl. 394, 61-73, 2012. https://doi.org/10.1016/j.jmaa.2012.04.065.Pólya, G. and Szegö, G. Problems and Theorems in Analysis, Vol. 1. New York: Springer-Verlag, 1997.Ramanujan, S. Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar, and B. M. Wilson). Providence, RI: Amer. Math. Soc., p. 327, 2000.Ramanujan, S. Question No. 298. J. Indian Math. Soc. 1911.Sizer, W. S. "Continued Roots." Math. Mag. 59, 23-27, 1986.Vieta, F. Uriorum de rebus mathematicis responsorum. Liber VII. 1593. Reprinted in New York: Georg Olms, pp. 398-400 and 436-446, 1970.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.Wong, B. and McGuffin, M. "The Museum of Infinite Nested Radicals." https://www.dgp.toronto.edu/~mjmcguff/math/nestedRadicals.html.

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Nested Radical

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Weisstein, Eric W. "Nested Radical." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NestedRadical.html

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