Trigonometry/Calculating Pi

From Wikibooks, open books for an open world
Jump to: navigation, search

Various formulae for calculating pi can be obtained from the power series expansion for .

Since , we have

This formula (due to Gottfried Leibniz) converges too slowly to be of practical use. However, similar formulae with much faster convergence can be found. John Machin (1680-1752) showed that


This formula was widely used by hand calculators. The first part of the right hand side is easy to calculate since finding Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \frac{1}{5^n}} involves very simple division, and the second part only needs 50 terms to compute 240 decimal places.

Leonhard Euler (1707-1783) showed that


Störmer showed that


and this formula was used in 1962 to calculate to over 100,000 decimals.