approximating sums of rational functions

Given a sum of the form m=nf(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 formPlanetmathPlanetmath. Furthermore, the approximation so obtained becomes better as n increases.

We begin with a simple illustrative example. Suppose that we want to sum m=n1/m2. We approximate m2 by m2-1/4, which factors as (m+1/2)(m-1/2). Then, upon separating the approximate summand into partial fractionsPlanetmathPlanetmath, the sum collapses:

m=n1(m+1/2)(m-1/2) =m=n(1m-1/2-1m+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