MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Define g(x)=exln(1+x).
(1) Write the first eight non-zero terms of the Maclaurin series for ex and the first eight non-zero terms for ln(1+x).
(2) Using the two series from part (a), determine—by collecting terms—the exact coefficient of x7 in the Maclaurin series for g(x). Clearly indicate each pair of terms whose product contributes to the x7 term.
(3) Verify that the coefficient you found in part (b) agrees with the coefficient obtained by differentiating your product series and evaluating at x=0. Provide a brief explanation of why the two approaches must yield the same value.