# approximating sums of rational functions

Given a sum of the form ${\sum}_{m=n}^{\mathrm{\infty}}f(m)$ 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
${\sum}_{m=n}^{\mathrm{\infty}}1/{m}^{2}$. 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:

$\sum _{m=n}^{\mathrm{\infty}}}{\displaystyle \frac{1}{(m+1/2)(m-1/2)}$ | $={\displaystyle \sum _{m=n}^{\mathrm{\infty}}}\left({\displaystyle \frac{1}{m-1/2}}-{\displaystyle \frac{1}{m+1/2}}\right)$ | ||

$={\displaystyle \sum _{m=n}^{\mathrm{\infty}}}{\displaystyle \frac{1}{m-1/2}}-{\displaystyle \sum _{m=n+1}^{\mathrm{\infty}}}{\displaystyle \frac{1}{m-1/2}}$ | |||

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

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

[general method to come]

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 |