approximating sums of rational functions

Given a sum of the form ${\sum }_{m=n}^{\mathrm{\infty }}f(m)$ 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 ${\sum }_{m=n}^{\mathrm{\infty }}1/m^2$. 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:

${\displaystyle \sum _{m=n}^{\mathrm{\infty }}}{\displaystyle \frac{1}{(m+1/2)(m-1/2)}}$ $={\displaystyle \sum _{m=n}^{\mathrm{\infty }}}\left({\displaystyle \frac{1}{m-1/2}}-{\displaystyle \frac{1}{m+1/2}}\right)$

$={\displaystyle \sum _{m=n}^{\mathrm{\infty }}}{\displaystyle \frac{1}{m-1/2}}-{\displaystyle \sum _{m=n+1}^{\mathrm{\infty }}}{\displaystyle \frac{1}{m-1/2}}$

$={\displaystyle \frac{1}{n-1/2}}$

Using a similar approach, we may estimate the error of our approximation.

[general method to come]

Title	approximating sums of rational functions
Canonical name	ApproximatingSumsOfRationalFunctions
Date of creation	2013-03-22 18:42:23
Last modified on	2013-03-22 18:42:23
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	7
Author	rspuzio (6075)
Entry type	Topic
Classification	msc 41A20