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

Solutions

April 16, 2003}
\end{center}

\begin{enumerate}

\item  \label{one}
For each of the power series below, determine the radius of convergence and the interval of convergence.
%
\[
\begin{array}{|c|c|c|}
\hline
\mbox{Series} & \mbox{Radius of Convergence} & \mbox{Interval of Convergence} \\
\hline && \\ 
\sum_{n=0}^\infty x^n & R=1 & I=(-1,1) \\
& & \\ \hline && \\
\sum_{n=0}^\infty nx^{n-1} & R=1 & I=(-1,1) \\
& & \\ \hline && \\
\sum_{n=0}^\infty \frac{x^{n+1}}{n+1} & R=1 & I=[-1,1) \\
& & \\ \hline && \\
\sum_{n=0}^\infty \frac{x^{n+2}}{(n+1)(n+2)} & R=1 & I=[-1,1] \\
& & \\ \hline 
\end{array}
\]

\item  Try and determine the relationship between the power series in question~\ref{one}.
	\\(Hint: write out the terms of the series.)

\inch

\item  Solve the initial value problem: $\frac{dy}{dx}=y$, $y(0)=1$.
	\\\sol
	This is a separation of variables.  First you solve and get $\ln|y| = x+C$ or $y=Ae^x$.
	Using the initial condition gives $y=e^x$.	

\end{enumerate}


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