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40-00-RaabesCriteria.tex
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40-00-RaabesCriteria.tex
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\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{RaabesCriteria}
\pmcreated{2013-03-22 15:08:45}
\pmmodified{2013-03-22 15:08:45}
\pmowner{pahio}{2872}
\pmmodifier{pahio}{2872}
\pmtitle{Raabe's criteria}
\pmrecord{7}{36892}
\pmprivacy{1}
\pmauthor{pahio}{2872}
\pmtype{Theorem}
\pmcomment{trigger rebuild}
\pmclassification{msc}{40-00}
\pmrelated{ASeriesRelatedToHarmonicSeries}
\endmetadata
% 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 here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
\begin{document}
\begin{thmplain}
\,The series\, $a_1\!+\!a_2\!+\!a_3\!+\cdots$\, with positive \PMlinkescapetext{terms} is
\begin{itemize}
\item convergent if, starting from some value of $n$, its \PMlinkescapetext{terms} fulfil the condition
$$\frac{a_{n+1}}{a_n} \leqq 1-\frac{\mu}{n}$$
where $\mu$ is a \PMlinkescapetext{constant} and $> 1$;
\item divergent if, starting from some value of $n$, its \PMlinkescapetext{terms} fulfil the condition
$$\frac{a_{n+1}}{a_n} \geqq 1-\frac{1}{n}-\frac{M}{n^2}$$
where $M$ is a \PMlinkescapetext{constant}.
\end{itemize}
\end{thmplain}
%%%%%
%%%%%
\end{document}