serie 1/(a^k-1)
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From: ksoileau on 20 Nov 2009 15:51 On Nov 18, 4:10 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 ? For large a, a good approximation is 1+1/a-ln(a-1)/ln(a)
From: Gottfried Helms on 20 Nov 2009 17:12 Am 20.11.2009 21:51 schrieb ksoileau: > On Nov 18, 4:10 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 ? 1/(a^1 -1) = 1/a + 1/a^2 + 1/a^3 + 1/a^4 + ... 1/(a^2 -1) = 1/a^2 + 1/a^4 + 1/a^6 + 1/a^8 + ... 1/(a^3 -1) = 1/a^3 + 1/a^6 + 1/a^9 + 1/a^12 +... 1/(a^4 -1) = 1/a^4 + 1/a^8 + 1/a^12 +1/a^16 + ... .... = .... ------------------------------------------------------ sum = 1/a + 2/a^2 + 2/a^3 + 3/a^4 + 2/a^5 + 4/a^6 + Here the numerators are the number-of-divisors of the exponent at a. I think I've seen some discussion of this in an article of Ed Sandifer about a work of L.Euler. Look for "Sandifer" "How Euler did it" at maa.online. Unfortunately I don't remember which of the monthly articles it was. hth - Gottfried Helms
From: Gerry Myerson on 22 Nov 2009 17:25 In article <7moii2F3id7vlU1(a)mid.dfncis.de>, Gottfried Helms <helms(a)uni-kassel.de> wrote: > Am 20.11.2009 21:51 schrieb ksoileau: > > On Nov 18, 4:10 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 ? > > 1/(a^1 -1) = 1/a + 1/a^2 + 1/a^3 + 1/a^4 + ... > 1/(a^2 -1) = 1/a^2 + 1/a^4 + 1/a^6 + 1/a^8 + ... > 1/(a^3 -1) = 1/a^3 + 1/a^6 + 1/a^9 + 1/a^12 +... > 1/(a^4 -1) = 1/a^4 + 1/a^8 + 1/a^12 +1/a^16 + ... > ... = .... > ------------------------------------------------------ > sum = 1/a + 2/a^2 + 2/a^3 + 3/a^4 + 2/a^5 + 4/a^6 + > > Here the numerators are the number-of-divisors of the exponent at a. > > I think I've seen some discussion of this in an article of > Ed Sandifer about a work of L.Euler. Look for "Sandifer" > "How Euler did it" at maa.online. Unfortunately I don't > remember which of the monthly articles it was. Peter Borwein proved that if r is rational and q is an integer, q > 1, then sum 1 / (q^n + r) is irrational. The reference is On the Irrationality of sum 1 / (q^n + r), J Number Theory 37 (1991) 253-259. I know that doesn't answer the question about closed form, but still that paper might be a good place to look for information. -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Raymond Manzoni on 22 Nov 2009 18:34
Gerry Myerson a écrit : > In article <7moii2F3id7vlU1(a)mid.dfncis.de>, > Gottfried Helms <helms(a)uni-kassel.de> wrote: > >> Am 20.11.2009 21:51 schrieb ksoileau: >>> On Nov 18, 4:10 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 ? >> 1/(a^1 -1) = 1/a + 1/a^2 + 1/a^3 + 1/a^4 + ... >> 1/(a^2 -1) = 1/a^2 + 1/a^4 + 1/a^6 + 1/a^8 + ... >> 1/(a^3 -1) = 1/a^3 + 1/a^6 + 1/a^9 + 1/a^12 +... >> 1/(a^4 -1) = 1/a^4 + 1/a^8 + 1/a^12 +1/a^16 + ... >> ... = .... >> ------------------------------------------------------ >> sum = 1/a + 2/a^2 + 2/a^3 + 3/a^4 + 2/a^5 + 4/a^6 + >> >> Here the numerators are the number-of-divisors of the exponent at a. >> >> I think I've seen some discussion of this in an article of >> Ed Sandifer about a work of L.Euler. Look for "Sandifer" >> "How Euler did it" at maa.online. Unfortunately I don't >> remember which of the monthly articles it was. > > Peter Borwein proved that if r is rational and q is an integer, > q > 1, then sum 1 / (q^n + r) is irrational. The reference is > On the Irrationality of sum 1 / (q^n + r), J Number Theory 37 > (1991) 253-259. I know that doesn't answer the question about > closed form, but still that paper might be a good place to look > for information. > A search on "Lambert series" could help at least to name this sum! : <http://en.wikipedia.org/wiki/Lambert_series> as well as the irrationality proof of Erdös : <http://www.math-inst.hu/~p_erdos/1948-04.pdf> for a=2 you get the 'Erdös–Borwein constant' : <http://en.wikipedia.org/wiki/Erdős–Borwein_constant> a later paper of Borwein : <http://www.mathaware.org/proc/1999-127-06/S0002-9939-99-04722-X/S0002-9939-99-04722-X.pdf> for recent results see Tachiya(2004) : <http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/25757/1/1384-34.pdf> and Matala-Aho(2006) 'New irrationality measures for q-logarithms' <http://cc.oulu.fi/~tma/TAPANI21.pdf> Hoping it helped, Raymond |