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Paris Constant

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

phi=sqrt(1+sqrt(1+sqrt(1+sqrt(1+...)))),
(1)

which can be written recursively as

phi_n=sqrt(1+phi_(n-1))
(2)

for n>=2, with phi_1=1.

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

phi-phi_n∼(2C)/((2phi)^n)
(3)

as n->infty, where

C=1.0986419643...
(4)

(OEIS A105415) is the Paris constant.

A product formula for C is given by

C=product_(n=2)^infty(2phi)/(phi+phi_n)
(5)

(Finch 2003, p. 8).

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

F(x)=2phiF(phi-sqrt(phi^2-x))
(6)

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

C=phiF(1/phi)
(7)

(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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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." http://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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Paris Constant

Cite this as:

Pegg, Ed Jr. and Weisstein, Eric W. "Paris Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ParisConstant.html

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