are called nested radicals. Herschfeld (1935) proved that a nested radical of realnonnegative 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.

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

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