limit comparison testLimitComparisonTest2005-02-10 07:44:582005-02-10 07:44:58Theorem% 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{amsthm}
\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
\newtheorem{thm}{Theorem}
\newtheorem{defn}{Definition}
\newtheorem{prop}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}
% Some sets
\newcommand{\Nats}{\mathbb{N}}
\newcommand{\Ints}{\mathbb{Z}}
\newcommand{\Reals}{\mathbb{R}}
\newcommand{\Complex}{\mathbb{C}}
\newcommand{\Rats}{\mathbb{Q}}The following theorem is a powerful test for convergence of series.
\begin{thm}[Limit \PMlinkescapetext{Comparison Test}] Let $\sum_{n=0}^\infty a_n$ and
$\sum_{n=0}^\infty b_n$ be two series of positive numbers. \\
\begin{enumerate}
\item If the limit $$\lim_{n\to \infty} \frac{a_n}{b_n}=L$$ exists and $L\neq 0$ is a
non-zero finite number, then both series $\sum_{n=0}^\infty a_n$ and
$\sum_{n=0}^\infty b_n$ converge or both diverge.\\
\item If $L=0$ and $\sum_{n=0}^\infty b_n$ converges then $\sum_{n=0}^\infty a_n$ converges as well. If $L=0$ and $\sum_{n=0}^\infty a_n$ diverges then $\sum_{n=0}^\infty b_n$ diverges as well.\\
\item Similarly, if the limit is infinite (``$L=\infty$'') and $\sum_{n=0}^\infty a_n$ converges then $\sum_{n=0}^\infty b_n$ converges as well. If $L=\infty$ and $\sum_{n=0}^\infty b_n$ diverges then $\sum_{n=0}^\infty a_n$ diverges as well.
\end{enumerate}
\end{thm}