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:

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

[general method to come]