Tea Time Numerical Analysis · Problem 1.2.19

Preliminaries — Leon Brin

Mathematics — Numerical Analysis · maclaurin-series

Problem

Exercise 18 details a rather inefficient means of obtaining an approximation to

\pi

. The method can be improved substantially by observing that

\pi/4 = \arctan \frac{1}{2} + \arctan \frac{1}{3}

and evaluating the series for the arctangent at

\frac{1}{2}

and at

\frac{1}{3}

. Determine the number of terms that must be summed to ensure an approximation to

\pi

within

10^{-3}