Nested Radical -- from Wolfram MathWorld

Calculus and Analysis > Special Functions > Radicals >

Nested Radical

Expressions of the form

(1)

are called nested radicals. 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/pi=sqrt(1/2)sqrt(1/2+1/2sqrt(1/2))sqrt(1/2+1/2sqrt(1/2+1/2sqrt(1/2)))...

(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)

Nest 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)

The silver constant is related to the nested radical expression

(12)

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

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

(13)

which gives as special cases

(14)

(, , ),

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

(15)

(), and

(16)

(). Equation also (13) gives rise to

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

(17)

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))+...))))).

(18)

Equation (◇) can be generalized to

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

(19)

for integers , which follows from

			(20)
			(21)
			(22)
			(23)
			(24)

In particular, taking gives

(25)

(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(...))^_^_,

(26)

which gives the special cases

(27)

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

3=sqrt(1+2sqrt(1+3sqrt(1+4sqrt(1+5sqrt(...)))))

(28)

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

For a nested radical of the form

(29)

to be equal a given real number , it must be true that

(30)

(31)

and

(32)

An amusing nested radical follows rewriting the series for e

(33)

(34)

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

(35)

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

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, 1992.

Landau, S. "Simplification of Nested Radicals." SIAM J. Comput. 21, 85-110, 1992.

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.

Pólya, G. and Szegö, G. Problems and Theorems in Analysis, Vol. 1. New York: Springer-Verlag, 1997.

Ramanujan, S. Question No. 298. J. Indian Math. Soc. 1911.

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.

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." http://www.dgp.toronto.edu/~mjmcguff/math/nestedRadicals.html.

CITE THIS AS:

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