MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Consider the function g(x)=arctanx for ∣x∣<1.
(a) Derive the Maclaurin series for 1+x21 and state its interval of convergence.
(b) Integrate the series you obtained in part (a) term by term from 0 to x to obtain a Maclaurin series for g(x). Write the first four non-zero terms explicitly.
(c) Use the series from part (b) to approximate arctan(0.3) with the first three non-zero terms. Estimate the absolute error of your approximation by applying an appropriate remainder estimate and state whether the required accuracy is within 5×10−6. Show all work clearly.