Trigonometry/Calculating Pi

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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(\tfrac{1}{5}\right)-\arctan\left(\tfrac{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(\tfrac{1}{7}\right)+2\arctan\left(\tfrac{3}{79}\right) .

Störmer showed that

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

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