Mathematics

Problem

Let

h(x)=\int_{0}^{x}\left(\int_{0}^{u}\sin t\,dt\right)du.

Assume

|x|<\infty

. (a) Starting with the Maclaurin series for

\sin t

, find a power-series representation for

\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)

and state its general term. (c) Determine the radius of convergence of the series for

h(x)

. (d) Approximate

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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