Golden Mean Series

Golden Mean Series

This page documents the development of an infinite series describing one of the classical numerical constants, Phi  or   , known as the Golden Mean or Golden Ratio.

Background:  Historically, four of the most famous irrational mathematical constants are , , , and .  Most of these have well documented and explicit representations in the form of infinite series and continued fractions (see table below for a summary of these representations). While investigating the Golden Mean, no equivalent infinite series was found that explicitely defined .

Classical Constant


Infinite Series Representation


Continued Fraction Representation


Archimedes' constant;             



Golden Mean;                             




Natural logarithmic base;     





Pythagoras' constant;         

New Series: A Taylor series expansion approach was used to define an infinite series that explicitly defines    without the use of transcendental functions. The resultant series that completes the table above is detailed here.


  Although the infinite series presented here for Phi, as well as the other series in the table above, are not necessarily the fastest way to calculate many decimal places for irrational numbers, the equations here are a straight forward way to calculate out the value of these constants using Excel or other symbolic calculation programs like Mathematica.

Brian Roselle