integral test IntegralTest 2002-02-24 08:02:36 2004-03-11 17:51:34 Theorem \usepackage{graphicx} %\usepackage{xypic} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand{\mathbb}[1]{\mathbbmss{#1}} \newcommand{\figura}[1]{\begin{center}\includegraphics{#1}\end{center}} \newcommand{\figuraex}[2]{\begin{center}\includegraphics[#2]{#1}\end{center}} \newtheorem{dfn}{Definition} Consider a sequence $(a_n)=\{a_0,a_1,a_2,a_3,\ldots\}$ and given $M\in \R$ consider any monotonically nonincreasing function $f:[M,+\infty)\to \R$ which extends the sequence, i.e. \[ f(n) = a_n \qquad \forall n\ge M \] An example is $$a_n = 2n\qquad \to\qquad f(x) = 2x$$ (the former being the sequence $\{0,2,4,6,8,\ldots\}$ and the later the doubling function for any real number. We are interested on finding out when the summation $$\sum_{n = 0}^{\infty}a_n$$ converges. The integral test states the following. The series $$\sum_{n = 0}^{\infty}a_n$$ converges if and only if the integral $$\int_M^\infty f(x)\, dx$$ is finite.