Trigonometry/Calculating Pi

Various formulae for calculating pi can be obtained from the power series expansion for $\arctan (x)$ .

Since $\arctan (1)=\frac{\pi }{4}$ , we have

$\frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\cdots$

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

$\frac{\pi }{4}=4\arctan \left(\frac{1}{5}\right)-\arctan \left(\frac{1}{239}\right)$ .

This formula was widely used by hand calculators. The first part of the right hand side is easy to calculate since finding $\frac{1}{5^n}$ involves very simple division, and the second part only needs 50 terms to compute 240 decimal places.

Leonhard Euler (1707-1783) showed that

$\frac{\pi }{4}=5\arctan \left(\frac{1}{7}\right)+2\arctan \left(\frac{3}{79}\right)$ .

Störmer showed that

$\frac{\pi }{4}=6\arctan \left(\frac{1}{8}\right)+2\arctan \left(\frac{1}{57}\right)+\arctan \left(\frac{1}{239}\right)$ ,

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