40A05-ComparisonTest.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{ComparisonTest}
\pmcreated{2013-03-22 13:21:48}
\pmmodified{2013-03-22 13:21:48}
\pmowner{mathwizard}{128}
\pmmodifier{mathwizard}{128}
\pmtitle{comparison test}
\pmrecord{4}{33889}
\pmprivacy{1}
\pmauthor{mathwizard}{128}
\pmtype{Theorem}
\pmcomment{trigger rebuild}
\pmclassification{msc}{40A05}

\endmetadata

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\begin{document}
The series
$$\sum_{i=0}^\infty a_i$$
with real $a_i$ is absolutely convergent if there is a sequence $(b_n)_{n\in\mathbb{N}}$ with positive real $b_n$ such that
$$\sum_{i=0}^\infty b_i$$
is \PMlinkescapetext{convergent} and for all sufficiently large $k$ holds $|a_k|\leq b_k$.

Also, the series $\sum a_i$ is divergent if there is a sequence $(b_n)$ with positive real $b_n$, so that $\sum b_i$ is divergent and $a_k\geq b_k$ for all sufficiently large $k$.
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\end{document}