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CTK Exchange
Alan
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Apr-12-07, 06:01 AM (EST) |
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"infinite sum over rational function"
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Is there any sort of general method for evaluating an infinite sum of an arbitrary rational function? For example, sum 1/n diverges, but the alternating harmonic series converges to log(2). Combining each pair of terms of the alternating harmonic series, you can rewrite the sum as 1/(2n*(2n-1)). Is there some direct way to show that this sum converges to log(2)? For another example, the sum 1+1/3-1/2 + 1/5+1/7-1/4 + 1/9+1/11-1/6 also converges, to log(8)/2. This sum can be written as a rational function with a first order numerator and a third order denominator. How can this be evaluated? Thanks. |
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JJ
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May-09-07, 01:20 AM (EST) |
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2. "RE: infinite sum over rational function"
In response to message #1
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A general method for evaluating infinite sums of arbitrary rational function is known, using the polygamma functions.
For example, see a rundown in "Summation of rational Series by Means of Polygamma Functions", Section 6.8 in "Handbook of Mathenatical Functions", by M.Abramowitz and I.A.Stegun, Dover publications, N.-Y., 9th.edit., 1970. |
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