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

This can be written recursively as

for , with .

Paris (1987) proved approached at a constant rate, namely

as , where

(Sloane's 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 , subject to initial conditions and . Then

(Finch 2003, p. 8).

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

 

Golden Ratio, Nested Radical