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\centerline{\bf Power Series Coefficients}
\medskip

Consider the power series
$$
f(z) = \sum_{n=0}^{\infty} c_{n}\left(z-z_{0}\right)^{n}
$$
expanded around the point $z_{0}$. In lecture we derived that the coeffcients are given by
$$
c_{n} = \frac{1}{n!}f^{(n)}(z_{0})
$$
\begin{enumerate}
\item Find the first four nonzero coeffcients for $\sin{\theta}$ expanded around the origin.

\vfill
\item Write out the series approximation, correct to fourth order, for $\sin{\theta}$ expanded around the
origin.

\bigskip\bigskip
\centerline{$\sin{\theta}$ = \underline{\hspace{4 in}}}
\bigskip\bigskip

\item Find the first four nonzero coe±cients for $\sin{\theta}$ expanded around $\theta_{0} = \frac{\pi}{6}$.

\vfill
\item Write out the series approximation, correct to fourth order, for $\sin{\theta}$ expanded around $\theta_{0} = \frac{\pi}{6}$.

\bigskip\bigskip
\centerline{$\sin{\theta}$ = \underline{\hspace{4 in}}}
\bigskip\bigskip
\item What does it mean to write a series expansion around the point a?

\vfill
\item Briefly describe in words how to expand the series approximation for a function, correct to fourth order.
\end{enumerate}

\vfill
\leftline{\it by Corinne Manogue}
\leftline{\copyright 2005 Corinne A. Manogue}
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