Expressions of the form
(1)
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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)
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(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)
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(4)
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(5)
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(6)
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Nest radicals also appear in the computation of the golden ratio
(7)
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and plastic constant
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Both of these are special cases of
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which can be exponentiated to give
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so solutions are
(11)
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In particular, for , this gives
(12)
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The silver constant is related to the nested radical expression
(13)
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There are a number of general formula for nested radicals (Wong and McGuffin). For example,
(14)
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which gives as special cases
(15)
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(, , ),
(16)
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(), and
(17)
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(). Equation (14) also gives rise to
(18)
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which gives the special case for , , , and ,
(19)
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Equation (◇) can be generalized to
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for integers , which follows from
(21)
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(22)
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(24)
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In particular, taking gives
(26)
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(J. R. Fielding, pers. comm., Oct. 8, 2002).
Ramanujan discovered
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which gives the special cases
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for , (Ramanujan 1911; Ramanujan 2000, p. 323; Pickover 2002, p. 310), and
(29)
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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)
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as
(31)
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so
(32)
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(J. R. Fielding, pers. comm., May 15, 2002).