MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Define q(x)=1−3x1. Let S(x) denote the power series for q(x) about x=0.
(a) Write S(x) in sigma notation and list its first three non-zero terms.
(b) Obtain a power-series representation for q′(x) by differentiating S(x) term-by-term. Write the result in sigma notation and include the first three non-zero terms explicitly.
(c) Use the series from part (b) to determine the exact coefficient of x7 in the Maclaurin series of q′(x).
(d) Verify that the series obtained for q′(x) converges on the same interval as S(x), and explain why term-by-term differentiation is justified within this interval.