MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Consider the function q(x)=xln(1+x), defined for x>−1 with x=0.
(a) Starting with the Maclaurin series for ln(1+x), obtain the Maclaurin series for q(x) up to and including the term in x3.
(b) Use the series from part (a) to evaluate x→0limq(x).
(c) At x=0.2, approximate q(0.2) using the polynomial found in part (a). Using the alternating-series remainder or another appropriate error estimate, bound the magnitude of the truncation error when the series is cut off after the x3 term.