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

April 2, 2003}
\end{center}

\textbf{Series Toolbox: }

\begin{enumerate}[I.]
\item  If $a_n$ is a geometric sequence, then $\sum_{n=1}^{\infty} a_n$ converges if $|r|<1$ and diverges if $|r| \geq 1$.  If $|r|<1$, then $\sum_{n=1}^{\infty} a_n = \frac{a}{1-r}$ where $a$ is the first term and $r$ is the common ratio.

\item  If $\lim_{n \ra \infty} a_n \neq 0$ then $\sum_{n=1}^{\infty} a_n$ diverges.

\item  If $\lim_{n \ra \infty} a_n = 0$ then $\sum_{n=1}^{\infty} a_n$ might diverge or it might converge (thing about $\lim_{n \ra \infty} \frac{1}{n}$).

\end{enumerate}



\begin{enumerate}

\item  Determine of the following series diverge or converge.  If the series converges, try to find the sum.
	\begin{enumerate}
	\item  $\sum_{n=1}^\infty 5^{-n}$
\bigskip

	\item  $\sum_{n=0}^\infty 5^{-n}$
\bigskip

	\item  $\sum_{n=1}^{\infty} \sqrt[n]{2}$
\bigskip

	\item  $\sum_{n=1}^{\infty} \frac{1+2^n}{5^n}$
\bigskip

	\item  $\sum_{n=0}^\infty \sin^n 1$
\bigskip

	\end{enumerate}


\item  
	\begin{enumerate}
	\item  Suppose you know that $\sum_{n=1}^\infty a_n$ converges.  How about $\sum_{n=100}^\infty a_n$?  Why or why not?
\bigskip

	\item  Suppose you know that $\sum_{n=1}^\infty a_n$ diverges.  How about $\sum_{n=100}^\infty a_n$?  Why or why not?
\bigskip
	\end{enumerate}

\newpage
\lecture

\item  Use the integral test to determine if the following series converge or diverge
	\begin{enumerate}
	\item  $\sum_{k=0}^{\infty} \frac{1}{k^2+1}$
\bigskip

	\item  $\sum_{n=1}^{\infty} \frac{4}{4n-1}$
\bigskip
	\end{enumerate}

\item  Use the comparison test to determine if the following series converge or diverge
	\begin{enumerate}
	\item  $\sum_{k=0}^{\infty} \frac{1}{k^2+1}$
\bigskip

	\item  $\sum_{n=1}^\infty \frac{4}{4n-1}$
\bigskip
	\end{enumerate}

\end{enumerate}


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