Mathematics

Problem

The Maclaurin series for

\sin x

\sin x = \sum_{n=0}^{\infty} \frac{(-1)^{n}x^{2n+1}}{(2n+1)!}.

Consider the function

p(x)=\sin(x^{2})

. (a) Obtain a power series representation for

p(x)

centered at

x=0

by making an appropriate substitution in the series above. (b) Using your series, determine the exact value of

p^{(18)}(0)

. (c) State the radius of convergence of the series for

p(x)

and justify your answer.