Mathematics

Problem

The Maclaurin series for

\arctan x

x-\dfrac{x^{3}}{3}+\dfrac{x^{5}}{5}-\dfrac{x^{7}}{7}+\dots

. When this series is used to approximate

\arctan\left(\dfrac13\right)

with its first four nonzero terms, calculate the maximum possible error of the approximation.