The Paris constant characterizes the rate of convergence of successive truncations of the nested radical
for the golden ratio. 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 that 
approaches 
geometrically, with asymptotic error
 |
(3)
|
as 
, where
 |
(4)
|
(OEIS A105415) is the Paris constant.
An infinite product for 
is given by
 |
(5)
|
(Finch 2003, p. 8).
Another formula is given by letting 
be the analytic function
satisfying 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.).
