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\begin{center}
{\Large Warm-Up Problems and Lecture Problems

April 18, 2003}
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Power series that we know:
\begin{eqnarray*}
\frac{1}{1-x} & = & \sum_{n=0}^\infty x^n = 1+x+x^2+x^3+ \cdots \\
\ln(1-x) & = & - \sum_{n=1}^\infty \frac{x^n}{n} = -x-\frac{x^2}{2} - \frac{x}{3}-\frac{x}{4}-\cdots \\
e^x & = & \sum_{n=0}^\infty \frac{x^n}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots \\
\tan^{-1}x & = & \sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{2n+1} = x-\frac{x^3}{3}+\frac{x^5}{5} - \cdots
\end{eqnarray*}


\begin{enumerate}

\item 
	\begin{enumerate}
	\item  Find the derivative of $\frac{1}{1-x}$.

	\item  Find a power series for $\frac{1}{(1-x)^2}$.
	\end{enumerate}
\inch


\item  Find a power series for $e^{2x}$.
\inch

\item  Find a power series, centered at $x=0$, for $\frac{1}{4+x}$
\inch

\item  Find a power series, centered at $x=0$, for $\ln(4+x)$. 
	

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\lecture

\item  Find the Taylor series for $\sin x$, centered at $x=\frac{\pi}{6}$.


\end{enumerate}


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