Chapter 4, Problem 4P

The Maclaurin series expansion for the arctangent of x is defined for $\left|x\right|\le 1$ as

arctan $x={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{{\left(-1\right)}^n}{2n+1}{x}^{2x+1}}$

(a) Write out the first four terms $\left(n=0,\mathrm{...},3\right)$.

(b) Starting with the simplest version, arctan $x=x$, add terms one at a time to estimate arctan $\left(\pi /6\right)$. After each new term is added, compute the true and approximate percent relative errors. Use your calculator to determine the true value. Add terms until the absolute value of the approximate error estimate falls below an error criterion conforming to two significant figures.