Let h(x)= _ 0 ^ x ( _ 0 ^ u t ,dt )du

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Let h(x)=∫0x(∫0usin⁡t dt)du.h(x)=\int_{0}^{x}\left(\int_{0}^{u}\sin t\,dt\right)du. Assume ∣x∣<∞|x|<\infty. (a) Starting with the Maclaurin series for sin⁡t\sin t, find a power-series representation for ∫0usin⁡t dt\displaystyle \int_{0}^{u}\sin t\,dt. Write the first four non-zero terms. (b) Use the result from part (a) to write a Maclaurin series for h(x)h(x) and state its general term. (c) Determine the radius of convergence of the series for h(x)h(x). (d) Approximate h(0.5)h(0.5) by using the first four non-zero terms of the series found in part (b). Provide an error estimate by comparing with the magnitude of the next term in the series.

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