Mathematics

Problem

The Maclaurin series for

\arctan x

\arctan x = \sum_{n=0}^{\infty} \frac{(-1)^{n}x^{2n+1}}{2n+1}\quad (|x|<1).

Let

g(x)=\dfrac{1}{1+x^{2}}

. (a) Determine a power series representation for

g(x)

centered at

x=0

and state its interval of convergence. (b) Using the first three non-zero terms of your series, approximate

g(0.2)

. Then provide an upper bound for the magnitude of the error in your approximation.