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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