ProofOfAbelsTestForConvergence 2002-12-27 05:43:06 2002-12-27 05:43:06 Proof 0 % 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 Let $b$ be the limit of $\{b_n\}$ and let $d_n=b_n-b$ when $\{b_n\}$ is decreasing and $d_n=b-b_n$ when $\{b_n\}$ is increasing. By Dirichlet's convergence test, $\sum a_nd_n$ is convergent and so is $\sum a_nb_n = \sum a_n(b\pm d_n) = b\sum a_n \pm \sum a_nd_n$.