Chapter 1.3, Problem 3ES

The Maclaurin series for the arctangent function converges for −1 < x ≤ 1 and is given by

$\arctan x=\underset{n\to \infty }{\lim }{P}_n(x)=\underset{n\to \infty }{\lim }{\displaystyle \underset{i=1}{\overset{n}{\sum }}{(-1)}^{i+1}}\frac{x^{2i-1}}{2i-1}.$

1. a. Use the fact that tan π/4 = 1 to determine the number of n terms of the series that need to be summed to ensure that |4P_n(1) − π| < 10⁻³.
2. b. The C++ programming language requires the value of π to be within 10⁻¹⁰. How many terms of the series would we need to sum to obtain this degree of accuracy?