Tea Time Numerical Analysis · Problem 1.2.18

Preliminaries — Leon Brin

Mathematics — Numerical Analysis · maclaurin-series

Problem

The Maclaurin series for the arctangent function converges for

-1 < x \leq 1

and is given by

\arctan x = \lim_{n\to\infty} P_n(x) = \lim_{n\to\infty} \sum_{i=1}^{n} (-1)^{i-1} \frac{x^{2i-1}}{2i-1}.

Use the fact that

\tan(\pi/4) = 1

to determine the number of terms,

n

, of the series that need to be summed to ensure that

|4P_n(1) - \pi| < 10^{-3}