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\title{Math 1b.  Lecture 19 \\
Other Convergence Tests 
(The Ratio Test)}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To understand and be able to apply the ration test to determine the convergence or divergence of a series.




\end{itemize} 

\section{The Ratio Test}

Given the series $\sum_{n=1}^\infty a_n$, let
$$
L = \lim_{n \to \infty} \left|  \frac{a_{n+1}}{a_n} \right|.
$$
\begin{itemize}

\item 
If $L < 1$, then the series $\sum a_n$ is absolutely convergent.

\item 
If $L > 1$, then the series $\sum a_n$ is diverges.

\item 
If $L =1$, the {\em Ratio Test\/} fails.

\end{itemize}



\section{Worksheet Problems}

\begin{enumerate}

\item
Determine the convergence of $\displaystyle \sum_{n=1}^\infty \frac{n!}{(2n)!}$.

\item
Determine the convergence of $\displaystyle\sum_{n=1}^\infty \frac{n!}{(n+2)!}$.

\item
Determine the convergence of $\displaystyle\sum_{n=1}^\infty \frac{1}{\sqrt[3]{n^2 - 1}}$.

\item
Determine the convergence of $\displaystyle\sum_{n=1}^\infty \frac{\arctan n}{1 + n^2}$.

\item
Determine the convergence of $\displaystyle\sum_{n=1}^\infty \frac{n^3}{n^5 + 4}$.

\item
Determine the convergence of $\displaystyle\sum_{n=1}^\infty \frac{1}{n 3^n}$.




\item
Determine the convergence of $\displaystyle1 + \frac{1}{1 \cdot 3} + \frac{1}{1 \cdot 3 \cdot 5} + \frac{1}{1 \cdot 3 \cdot 5 \cdot 7} + \cdots $.


\end{enumerate}





\section*{References}

\begin{itemize}

\item
\S 8.4 in 
James Stewart.
{\it Single Variable Calculus: Concepts \& Context}, third edition.
Brooks/Cole, Belmont CA, 2005.
ISBN 0--534--41022--7.


\end{itemize}



\section*{Notes}

\subsection*{March 24, 2006}



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