Various formulae for calculating pi can be obtained from the power series expansion for tan-1(x).
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 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 π to over 100,000 decimals.