Paris Constant
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."
