MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Define H(x)=∫0xtsintdt. Answer the questions below.
(a) Write the Maclaurin series for sint and then obtain a power series for tsint by algebraic multiplication. State the general term clearly.
(b) Integrate the series from part (a) term by term from 0 to x to find a power-series representation for H(x). Give the first five non-zero terms.
(c) Evaluate H(x) in closed form without using the series and verify that your closed-form expression matches the sum from part (b).
(d) Determine the radius of convergence of the series for H(x) and discuss its convergence at x=π.