From: Boudewijn Moonen Subject: Re: An overlook method to sum a series Date: Mon, 25 Oct 1999 16:53:29 +0200 Newsgroups: sci.math To: fch Keywords: Euler-MacLaurin-summation technique fch wrote: > > Hi all, > > Here I have noticed a mathod that can be used to sum a series. > Surprisingly, it is not given anywhere in the past materials, so I like > anyone who is interested in summation techniques to look at it. Perhaps > drop some words. > > Click here to see: > http://www.alphalink.com.au/~fch/sum/index.html > > Have fun :> > fch@alphalink.com Presumably you are kicking a dead horse. This is the simplest case of a powerful, and *widely* known, summation technique known as "Euler-MacLaurin-summation"; so calling this "to have been overlooked" is what I call a true misnomer (Euler lived two and a halve centuries from now). Namely, consider your formula sum_{k=0}^{n-1} f(k) = int_0^n f(x) dx - sum_{k=0}^{n-1} ( int_0^n [f(x) - f(k)]dx ) (1) On the interval (k,k+1) the function g(x) := x - [x] - 1/2 is linear with slope 1, so has g'(x) = 1 there. Therfore, by partial integration int_0^n [f(x) - f(k)]dx = (x - [x] - 1/2)*(f(x) - f(k)|_k^{k+1} - - int_k^{k+1} (x - [x] - 1/2)*f'(x) Adding this up, the RHS telescopes, and plugging the result into (1) yields sum_{k=0}^{n-1} f(k) = (2) 1/2*(f(0) + f(n)) + int_0^n f(x) dx + int_0^n (x - [x] - 1/2)*f'(x) dx, the simplest case of the Euler-MacLaurin formula. For this and more refined formulas obtained by iterated partial integration, see F. Erwe, Differential- und Integralrechnung II, BI Mannheim 1962, in particular pp. 46-49. The formula (2) is (180) there. Somehow, to me it is a mixture of naivite, ignorance and arrogance to announce something which is widely known as "overlooked" without taking care of inquiring first. I think the better option is this case would have been to first modestly ask *whether* your summation technique was known or had been overlooked. BTW, to derive the geometric series in this way is cracking a peanut with a sledgehammer. How to derive the power sums by those methods, in fact for *general* k, has also been known for centuries, going back as far as to one of the Bernouillis. Regards, -- Boudewijn Moonen Institut fuer Photogrammetrie der Universitaet Bonn Nussallee 15 D-53115 Bonn GERMANY e-mail: Boudewijn.Moonen@ipb.uni-bonn.de Tel.: GERMANY +49-228-732910 Fax.: GERMANY +49-228-732712