Paris Constant

The golden ratio phi can be written in terms of a nested radical in the beautiful form


which can be written recursively as


for n>=2, with phi_1=1.

Paris (1987) proved phi_n approaches phi at a constant rate, namely


as n->infty, where


(OEIS A105415) is the Paris constant.

A product formula for C is given by


(Finch 2003, p. 8).

Another formula is given by letting F(x) be the analytic solution to the functional equation


for |x|<phi^2, subject to initial conditions F(0)=0 and F^'(0)=1. Then


(Finch 2003, p. 8).

A close approximation is ln3=1.09861..., which is good to 4 decimal places (M. Stark, pers. comm.).

See also

Golden Ratio, Nested Radical

Portions of this entry contributed by Ed Pegg, Jr. (author's link)

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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.", N. J. A. Sequence A105415 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Paris Constant

Cite this as:

Pegg, Ed Jr. and Weisstein, Eric W. "Paris Constant." From MathWorld--A Wolfram Web Resource.

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