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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Absolute Convergence and the Ratio and Root Tests">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{Absolute Convergence and the Ratio and Root Tests}

\begin{definition}
A series $\sum a_{n}$ is called \textbf{absolutely convergent} \label%
{absolutely convergent}if the series of absolute values $\sum \left|
a_{n}\right| $ is convergent.
\end{definition}

\begin{definition}
A series $\sum a_{n}$ is called \textbf{conditionally convergent} \label%
{conditionally convergent}if it is convergent but not absolutely convergent.
\end{definition}

\begin{theorem}
If a series $\sum a_{n}$ is absolutely convergent, then it is convergent.
\end{theorem}

\subsubsection{The Ratio Test\label{ratio test}}

\begin{enumerate}
\item  If $\lim\limits_{n\rightarrow \infty }\left| \frac{a_{n+1}}{a_{n}}%
\right| =L<1$, then the series $\sum\limits_{n=1}^{\infty }a_{n}$ is
absolutely convergent (and therefore convergent).

\item  If $\lim\limits_{n\rightarrow \infty }\left| \frac{a_{n+1}}{a_{n}}%
\right| =L>1$ or $\lim\limits_{n\rightarrow \infty }\left| \frac{a_{n+1}}{%
a_{n}}\right| =\infty $, then the series $\sum\limits_{n=1}^{\infty }a_{n}$
is divergent.
\end{enumerate}

\begin{remark}
If $\lim\limits_{n\rightarrow \infty }\left| \frac{a_{n+1}}{a_{n}}\right| =1$%
, the Ratio Test gives no information about the series. The series may
converge or diverge. Another test must be used in this case.
\end{remark}

\subsubsection{The Root Test\label{root test}}

\begin{enumerate}
\item  If $\lim\limits_{n\rightarrow \infty }\sqrt[n]{\left| a_{n}\right| }%
=L<1$, then the series $\sum\limits_{n=1}^{\infty }a_{n}$ is absolutely
convergent (and therefore convergent).

\item  If $\lim\limits_{n\rightarrow \infty }\sqrt[n]{\left| a_{n}\right| }%
=L>1$ or $\lim\limits_{n\rightarrow \infty }\sqrt[n]{\left| a_{n}\right| }%
=\infty $, then the series $\sum\limits_{n=1}^{\infty }a_{n}$ is divergent.
\end{enumerate}

\begin{remark}
If $\lim\limits_{n\rightarrow \infty }\sqrt[n]{\left| a_{n}\right| }=1$, the
Ratio Test gives no information about the series. The series may converge or
diverge. Another test must be used in this case.\bigskip 
\end{remark}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Ratio test}{}{}{../../Help/DM7-7.tex#Ratio test}}

{\small \hyperref{Root test}{}{}{../../Help/DM7-7.tex#Root test}}

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