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)

(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

(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,

(13)

which gives as special cases

(14)

(, , ),

(15)

(), and

(16)

(). Equation (13) also gives rise to

(17)

which gives the special case for , , , and ,

(18)

Equation (◇) can be generalized to

(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

(26)

which gives the special cases

(27)

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

(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)

so

(31)

and

(32)

An amusing nested radical follows rewriting the series for e

(33)

as

(34)

so

(35)

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

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