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\begin{document}

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

April 14, 2003}
\end{center}

\begin{enumerate}

\item  Find the domain of the following functions.
	(For which values of $x$ does the series converge?)
	\\(Use the ratio test and the remember to test the ``end points.''

	\begin{enumerate}
	\item  $f(x) = \sum_{n=0}^\infty x^n$
\inch

	\item  $g(x) = \sum_{n=1}^\infty \frac{(x-3)^n}{n}$
\inch

	\item  $h(x) = \sum_{n=0}^\infty \frac{x^n}{n!}$
\inch

	\item  $k(x) = \sum_{n=0}^\infty n!x^n$
\inch

	\item  $j(x) = \sum_{n=1}^\infty \frac{(x+2)^n}{2^n n^2}$

	\end{enumerate}


\newpage

\lecture

\item  For each of the power series below, determine where the series is centered, 
	and then determine the radius of convergence and the interval of convergence.
\[
\begin{array}{|c|c|c|c|}
\hline
\mbox{Series} & \mbox{Center of Power Series} & \mbox{Radius of Convergence} & \mbox{Interval of Convergence} \\
\hline &&& \\ 
\sum_{n=0}^\infty 3^nx^n & & & \\
& & & \\ \hline &&& \\
\sum_{n=0}^\infty 3^n(x-5)^n & & & \\
& & & \\ \hline &&& \\
\sum_{n=0}^\infty 3^n(2x-5)^n & & & \\
& & & \\ \hline &&& \\
\sum_{n=1}^\infty \frac{1}{n^3}(x+3)^n & & & \\
& & & \\ \hline &&& \\
\sum_{n=1}^\infty \frac{n}{(n+1)^2} (3x-4)^n & & & \\
& & & \\ \hline 
\end{array}
\]


\item  Find a power series representation for the following functions:
	\begin{enumerate}
	\item  $h(x) = \frac{x}{1-x}$.
\inch

	\item  $k(x) = \frac{x}{1-x^3}$
\inch

	\item  $g(x) = \frac{1}{2+x}$ 
		(Hint: you can rearrange $g$ to look like $g(x) = \frac{1}{2(1+x/2)}$
	\end{enumerate}

\end{enumerate}


\end{document}