ComparisonTest 2003-01-10 08:00:54 2003-01-10 08:00:54 Theorem % 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 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 $k$.