serie 1/(a^k-1)
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From: Patrick Coilland on 18 Nov 2009 05:10 Hello everybody Is there a known closed form for sum_{k=1,+infty}1/(a^k-1), where "a" is a positive integer > 1 ?
From: Dan Cass on 18 Nov 2009 00:21 > Hello everybody > > Is there a known closed form for > sum_{k=1,+infty}1/(a^k-1), where "a" is > a positive integer > 1 ? Maple doesn't know this sum in closed form. Maybe God knows it.
From: Patrick Coilland on 18 Nov 2009 11:01 Dan Cass a écrit : >> Hello everybody >> >> Is there a known closed form for >> sum_{k=1,+infty}1/(a^k-1), where "a" is >> a positive integer > 1 ? > > Maple doesn't know this sum in closed form. > Maybe God knows it. Ok, thanks :) I'll ask Him ;)
From: Jim Ferry on 18 Nov 2009 11:17 On Nov 18, 5:10 am, Patrick Coilland <pcoill...(a)pcc.fr> wrote: > Hello everybody > > Is there a known closed form for sum_{k=1,+infty}1/(a^k-1), where "a" is > a positive integer > 1 ? This can be expressed in terms of the q-digamma function, though this probably doesn't help much. See http://mathworld.wolfram.com/q-PolygammaFunction.html Let phi(z,q) denote the q-digamma function. Then sum_{k=1,+infty}1/(a^k-1) = 1 - (log(a-1) + phi(1,1/a))/log(a) for real a > 1. BTW, phi(1,q) increases from -EulerGamma to 0 as q decreases from 1 to 0.
From: Achava Nakhash, the Loving Snake on 18 Nov 2009 12:35
On Nov 18, 2:10 am, Patrick Coilland <pcoill...(a)pcc.fr> wrote: > Hello everybody > > Is there a known closed form for sum_{k=1,+infty}1/(a^k-1), where "a" is > a positive integer > 1 ? Just a little comment about English. The s on the end of a word usually means that it is a plural, but sometimes a word just ends in the letter s. One such word is series. We can have one series, or we can have two series, but there is no such word as serie in English. This is a common error in sci.math, and I am guessing at the reason it is made. At the moment I can't think of any other English words that end in s that aren't plural forms, but I expect some will come to me later. Just in case you care, Achava |