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# 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:

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

$\displaystyle=\sum_{{m=n}}^{\infty}{1\over m-1/2}-\sum_{{m=n+1}}^{\infty}{1% \over m-1/2}$ | |||

$\displaystyle={1\over n-1/2}$ |

[general method to come]

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