TelescopingSum 2004-06-17 10:56:37 2008-05-03 14:35:02 Definition telescope % 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 A {\em telescoping sum} is a sum in which cancellation occurs between subsequent terms, allowing the sum to be expressed using only the initial and final terms. Formally a telescoping sum is or can be rewritten in the form $$ S= \sum_{n=\alpha}^{\beta}\left(a_n - a_{n+1}\right) = a_\alpha - a_{\beta+1}$$ where $a_n$ is a sequence. {\bf Example:}\\ Define $S(N) = \sum_{n=1}^{N} \frac{1}{n(n+1)}$. Note that by partial fractions of expressions: \[ \frac{1}{n(n+1)}= \frac{1}{n} - \frac{1}{n+1} \] and thus $a_n = \frac{1}{n}$ in this example. \[ S(N) = \sum_{n=1}^{N} \left( \frac{1}{n} - \frac{1}{n+1} \right) \] \[ = \left( 1 - \frac{1}{2} \right) + \cdots + \left( \frac{1}{n} - \frac{1}{n+1} \right) + \left( \frac{1}{n+1} - \frac{1}{n+2} \right) + \cdots + \left( \frac{1}{N} - \frac{1}{N+1} \right) \] \[ = 1 + \left( - \frac{1}{2} + \frac{1}{2} \right) + \cdots + \left( -\frac{1}{n+1} +\frac{1}{n+1} \right) + \cdots - \frac{1}{N+1} \] \[ = 1 - \frac{1}{N+1} \]