ApproximatingSumsOfRationalFunctions 2009-01-06 15:11:22 2009-01-06 20:19:26 Topic % 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 Given a sum of the form $\sum_{m=n}^\infty f(m)$ where $f$ is a rational function, it is possible to approximate it by approximating $f$ by another rational function which can be summed in closed form. Furthermore, the approximation so obtained becomes better as $n$ increases. We begin with a simple illustrative example. Suppose that we want to sum $\sum_{m=n}^\infty 1/m^2$. We approximate $m^2$ by $m^2 - 1/4$, which factors as $(m+1/2)(m-1/2)$. Then, upon separating the approximate summand into partial fractions, the sum collapses: \begin{align*} \sum_{m=n}^\infty {1 \over (m+1/2)(m-1/2)} &= \sum_{m=n}^\infty \left( {1 \over m-1/2} - {1 \over m+1/2} \right) \\ &= \sum_{m=n}^\infty {1 \over m-1/2} - \sum_{m=n+1}^\infty {1 \over m-1/2} \\ &= {1 \over n-1/2} \end{align*} Using a similar approach, we may estimate the error of our approximation. [general method to come]