Here’s an interesting bit of history from Julian Havil’s new book The Irrationals. In 1593 Francois Vièta discovered the following infinite product for pi:
Havil says this is “the earliest known.” I don’t know whether this is specifically the oldest product representation for pi, or more generally the oldest formula for an infinite sequence of approximations that converge to pi. Vièta’s series is based on the double angle formula for cosine.
The first series for pi I remember seeing comes from evaluating the Taylor series for arc tangent at 1:
I saw this long before I knew what a Taylor series was. I imagine others have had the same experience because the series is fairly common in popular math books. However, this series is completely impractical for computing pi because it converges at a glacial pace. Vièta’s formula, on the other hand, converges fairly quickly. You could see for yourself by running the following Python code:
from math import sqrt prod = 1.0 radic = 0.0 for i in range(10): radic = sqrt(2.0 + radic) prod *= 0.5*radic print 2.0/prod
After 10 terms, Vièta’s formula is correct to five decimal places.
Posts on more sophisticated and efficient series for computing pi: