On Nov 18, 2:10 am, Patrick Coilland <pcoill...@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.
"Achava Nakhash, the Loving Snake" <ach...@hotmail.com> wrote:
> 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.
Kudos Biceps, triceps, ... Species (from taxonomy, rather than coined money), which reminds me of Homo sapiens
Those are a few that just happen to come to mind.
And of course, there are plural nouns in English which end in s but have no singular form. An example is
Pants (a garment, rather than what a dog does)
But one finds "pant" being used commonly in the clothing industry nowadays, and so I have little doubt that "pant" will be considered acceptable fairly soon.
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.
> 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 (ge...@maths.mq.edi.ai) (i -> u for email)
> In article <7moii2F3id7v...@mid.dfncis.de>, > Gottfried Helms <he...@uni-kassel.de> wrote:
>> Am 20.11.2009 21:51 schrieb ksoileau: >>> On Nov 18, 4:10 am, Patrick Coilland <pcoill...@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.
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