Re: [math-fun] Divergent sum = -1/12
The reason that people attribute -1/12 to this sum is that it can be considered to be Zeta[-1], where we define Zeta[z] (z > 1 for convergence) to be: 1/1^z + 1/2^z + 1/3^z + 1/4^z + ... Then Zeta[] has a unique analytic continuation to a meromorphic function over the complex plane (with a single pole at 1), and Zeta[-1] = -1/12. That's the only reasonable evidence I've seen in favour of this. The popular `numberphile' video giving a purported `proof' using elementary methods is unconvincing hand-wavery. Sincerely, Adam P. Goucher
----- Original Message ----- From: Bernie Cosell Sent: 04/16/14 07:10 PM To: math-fun Subject: [math-fun] Divergent sum = -1/12
I'm wondering what the solid [??] mathematical basis is for sum(n) = -1/12. I was looking at the series (which seems to be a starting place for the "proof" of -1/12) 1 - 1 + 1 - 1 .. and it can have any of a bunch of values, depending on how you look at it. {I vaguely recall from high school that you can sum it as (1-1) + (1-1) + (1-1), and so get the sum as being zero. OR you can sum it as 1 - (1 - 1) - (1 - 1) and get a sum as being 1. OR you can do the standard summation trick: S = 1 - 1 + 1 - 1 + 1 = 1 - S, ==> S = 1/2 and you can probably get it to "sum" to other values with other manipulations.
Are there some kind of [non algebraic?] extensions/definitions of 'sum' that are generally accepted for determining the "sums" of series that would appear not to have one.
/Bernie\
-- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--
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Well, it's Euler's proof, so not entirely unconvincing. Euler just had the exquisite taste to know which series would have analytic continuations even though the theory hadn't been developed yet. On Wed, Apr 16, 2014 at 4:12 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
The reason that people attribute -1/12 to this sum is that it can be considered to be Zeta[-1], where we define Zeta[z] (z > 1 for convergence) to be:
1/1^z + 1/2^z + 1/3^z + 1/4^z + ...
Then Zeta[] has a unique analytic continuation to a meromorphic function over the complex plane (with a single pole at 1), and Zeta[-1] = -1/12.
That's the only reasonable evidence I've seen in favour of this. The popular `numberphile' video giving a purported `proof' using elementary methods is unconvincing hand-wavery.
Sincerely,
Adam P. Goucher
----- Original Message ----- From: Bernie Cosell Sent: 04/16/14 07:10 PM To: math-fun Subject: [math-fun] Divergent sum = -1/12
I'm wondering what the solid [??] mathematical basis is for sum(n) = -1/12. I was looking at the series (which seems to be a starting place for the "proof" of -1/12) 1 - 1 + 1 - 1 .. and it can have any of a bunch of values, depending on how you look at it. {I vaguely recall from high school that you can sum it as (1-1) + (1-1) + (1-1), and so get the sum as being zero. OR you can sum it as 1 - (1 - 1) - (1 - 1) and get a sum as being 1. OR you can do the standard summation trick: S = 1 - 1 + 1 - 1 + 1 = 1 - S, ==> S = 1/2 and you can probably get it to "sum" to other values with other manipulations.
Are there some kind of [non algebraic?] extensions/definitions of 'sum' that are generally accepted for determining the "sums" of series that would appear not to have one.
/Bernie\
-- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
I have nothing but the highest respect for Euler, but he did not do rigorous math the way we mean that term today. So Euler's non-proof is not even faintly convincing to me in terms of the modern meaning of the convergence of a series. In terms of modern math notation, the equation (*) 1+2+3+... = -1/12 is true only if you change the meaning of the notation. On the other hand, Terry Tao (for whom I also have nothing but the greatest respect, and who uses modern math notation) has a lot to say about (*) here: < http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoul... >. --Dan On Apr 16, 2014, at 3:45 PM, Mike Stay <metaweta@gmail.com> wrote:
Well, it's Euler's proof, so not entirely unconvincing. Euler just had the exquisite taste to know which series would have analytic continuations even though the theory hadn't been developed yet.
On Wed, Apr 16, 2014 at 4:12 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
The reason that people attribute -1/12 to this sum is that it can be considered to be Zeta[-1], where we define Zeta[z] (z > 1 for convergence) to be:
1/1^z + 1/2^z + 1/3^z + 1/4^z + ...
Then Zeta[] has a unique analytic continuation to a meromorphic function over the complex plane (with a single pole at 1), and Zeta[-1] = -1/12.
That's the only reasonable evidence I've seen in favour of this. The popular `numberphile' video giving a purported `proof' using elementary methods is unconvincing hand-wavery.
participants (3)
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Adam P. Goucher -
Dan Asimov -
Mike Stay