approximating sums of rational functions

Given a sum of the form ${\sum }_{m=n}^{\mathrm{\infty }}f(m)$ where $f$ is a rationalfunction, it is possible to approximate it by approximating $f$ by anotherrational function which can be summed in closed form. Furthermore, theapproximation 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 $m^2-1/4$, whichfactors as $(m+1/2)(m-1/2)$. Then, upon separating the approximate summandinto 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]