% This is lecture no. 12 of MH3100/: Real Analysis I.

\chapter{Absolute Convergence Test}%
\label{cha:absolute_convergence_test}

\section{Absolute Convergence Test}%
\label{sec:absolute_convergence_test}

\begin{thrm}
	Absolute Convergence Test.
	If the series $\sum\abs{a_n}$ of \textbf{absolute values} is \textbf{convergent}, then the series $\sum\abs{a_n}$ is \textbf{convergent}.
\end{thrm}

\begin{proof}
	The Cauchy Criterion for the convergence of $\sum\abs{a_n}$ is
	\[
	\forall\varepsilon>0,\exists N\in\mathbb{N}\suchthat\forall n\in\mathbb{N}\quad n>N\implies \abs{\abs{a_{N+1}} + \abs{a_{N+2}} + \dotsb + \abs{a_n}}<\varepsilon
	.\]
	Therefore,
	\[
	\forall N,n\in\mathbb{N}\quad n>N \implies \abs{a_{N+1} + a_{N+2} + \dotsb + a_n}\leqslant\abs{a_{N+1}} + \abs{a_{N+2}} + \dotsb + \abs{a_n}
	\]
	shows that the Cauchy Criterion for the convergence of $\sum\abs{a_n}$ implies the Cauchy Criterion for the convergence of $\sum a_n$.
\end{proof}

\section{Absolute \& Conditional Convergence}%
\label{sec:absolute_&_conditional_convergence}

\begin{definition}
	Absolute \& Conditional Convergence.
	A series $\sum a_n$ is said to be \textbf{absolutely convergent} if the series $\sum\abs{a_n}$ converges.
	A series $\sum a_n$ is said to be \textbf{conditionally convergent} if it is convergent but not absolutely convergent.
\end{definition}

\begin{enumerate}
	\item The Absolute Convergence Test can be rephrased as
		\begin{quote}
			\emph{An absolutely convergent series is convergent.}
		\end{quote}
	\item The converse is not true in general; that is---there are conditionally convergent series.
	\item A series is either
		\begin{enumerate}
			\item absolutely convergent;
			\item conditionally convergent; or
			\item divergent.
		\end{enumerate}
\end{enumerate}
