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Nested Radical

lim_(k->infty)x_0+sqrt(x_1+sqrt(x_2+sqrt(...+x_k)))

(1)

are called nested radicals. Herschfeld (1935) proved that a nested radical ofreal nonnegative terms convergesiff (x_n)^(2^(-n)) is bounded. He also extended this result to arbitrarypowers (which include continued square roots andcontinued fractions as well), a result is known asHerschfeld's convergence theorem.

Nested radicals appear in the computation ofpi,

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), intrigonometrical values ofcosine andsine for argumentsof the form pi/2^n , e.g.,

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

Nest radicals also appear in the computation of thegoldenratio

(7)

andplastic constant

P=RadicalBox[{1, +, RadicalBox[{1, +, RadicalBox[{1, +, ...}, 3]}, 3]}, 3].

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Both of these are special cases of

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

(9)

which can be exponentiated to give

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

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so solutions are

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In particular, for n=2 , this gives

(12)

Thesilver constant is related to the nested radicalexpression

RadicalBox[{7, +, 7, RadicalBox[{7, +, ...}, 3]}, 3].

(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

(b+sqrt(b^2+4a))/2=sqrt(a+bsqrt(a+bsqrt(a+bsqrt(...))))

(15)

( n=2 , q=1-a/x^2 , x=b/q ),

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

(16)

( q=1 ), and

(17)

( q=1,n=2 ). 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 q=1/2 , n=2 , x=1 , and k=-1 ,

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 (◇) can be generalized to

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

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for integers n>=2 , which follows from

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

In particular, taking n=3 gives

sqrt(x)=RadicalBox[{x, RadicalBox[{x, RadicalBox[{x, ...}, 3]}, 3]}, 3].

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

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which gives the special cases

x+1=sqrt(1+xsqrt(1+(x+1)sqrt(1+(x+2)sqrt(1+...))))

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for a=0 , n=1 (Ramanujan 1911; Ramanujan 2000, p. 323; Pickover 2002, p. 310), and

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

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for a=0 , n=1 , and x=2 . The justification of this process in general (and in the particular example of lnsigma , where sigma isSomos's quadratic recurrence constant) is given by Vijayaraghavan (in Ramanujan 2000, p. 348).

An amusing nested radical follows rewriting the series fore

(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. Monthly98, 735-739, 1991.Herschfeld, A. "On Infinite Radicals."Amer. Math. Monthly42, 419-429, 1935.Jeffrey, D. J. and Rich, A. D. InComputer 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. " sqrt(2)+sqrt(3)

: 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.

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Nested Radical

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

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Nested Radical

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