40-00-RemainderTerm.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{RemainderTerm}
\pmcreated{2013-03-22 14:51:02}
\pmmodified{2013-03-22 14:51:02}
\pmowner{PrimeFan}{13766}
\pmmodifier{PrimeFan}{13766}
\pmtitle{remainder term}
\pmrecord{8}{36522}
\pmprivacy{1}
\pmauthor{PrimeFan}{13766}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{40-00}
\pmsynonym{remainder}{RemainderTerm}
\pmsynonym{tail of series}{RemainderTerm}
%\pmkeywords{partial sum}
\pmrelated{SumOfSeries}

\endmetadata

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\begin{document}
Let $S_n$ be the $n^\mathrm{th}$ partial sum of the series\, $a_1\!+\!a_2\!+\cdots$\, with real or complex \PMlinkescapetext{terms} $a_n$ ($n = 1,\,2,\,\ldots$). 
\begin{itemize}
  \item If the series is convergent with sum $S$, then we call the \PMlinkescapetext{difference}\, $R_n := S\!-\!S_n$\, the  $n^\mathrm{th}$ {\em remainder term} or simply {\em remainder} of the series ($n = 1,\,2,\,\ldots$).\, Then\, $\lim_{n\to\infty}R_n = 0$.
  \item If there exists a number $s$ such that\, $\lim_{n\to\infty}(s\!-\!S_n) = 0$,\, then the series is convergent and its sum is $s$.
\end{itemize}
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\end{document}