remainder termRemainderTerm2004-11-24 04:41:152006-09-29 02:48:04Definitionseriespartial sum% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands hereLet $S_n$ 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}