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approximating sums of rational functions
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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]
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"approximating sums of rational functions" is owned by rspuzio.
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Cross-references: estimate, similar, partial fractions, separating, factors, simple, approximation, closed form, rational function, sum
This is version 4 of approximating sums of rational functions, born on 2009-01-06, modified 2009-01-06.
Object id is 11470, canonical name is ApproximatingSumsOfRationalFunctions.
Accessed 200 times total.
Classification:
| AMS MSC: | 41A20 (Approximations and expansions :: Approximation by rational functions) |
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Pending Errata and Addenda
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