# approximating sums of rational functions

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:

 $\displaystyle\sum_{m=n}^{\infty}{1\over(m+1/2)(m-1/2)}$ $\displaystyle=\sum_{m=n}^{\infty}\left({1\over m-1/2}-{1\over m+1/2}\right)$ $\displaystyle=\sum_{m=n}^{\infty}{1\over m-1/2}-\sum_{m=n+1}^{\infty}{1\over m% -1/2}$ $\displaystyle={1\over n-1/2}$

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

[general method to come]

Title approximating sums of rational functions ApproximatingSumsOfRationalFunctions 2013-03-22 18:42:23 2013-03-22 18:42:23 rspuzio (6075) rspuzio (6075) 7 rspuzio (6075) Topic msc 41A20