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Date: 5-12-2020
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The golden ratio can be written in terms of a nested radical in the beautiful form
(1) |
which can be written recursively as
(2) |
for , with .
Paris (1987) proved approaches at a constant rate, namely
(3) |
as , where
(4) |
(OEIS A105415) is the Paris constant.
A product formula for is given by
(5) |
(Finch 2003, p. 8).
Another formula is given by letting be the analytic solution to the functional equation
(6) |
for , subject to initial conditions and . Then
(7) |
(Finch 2003, p. 8).
A close approximation is , which is good to 4 decimal places (M. Stark, pers. comm.).
REFERENCES:
Finch, S. R. "Analysis of a Radical Expansion." §1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 8, 2003.
Paris, R. B. "An Asymptotic Approximation Connected with the Golden Number." Amer. Math. Monthly 94, 272-278, 1987.
Plouffe, S. "The Paris Constant." https://pi.lacim.uqam.ca/piDATA/paris.txt.
Sloane, N. J. A. Sequence A105415 in "The On-Line Encyclopedia of Integer Sequences."
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