[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 <--
There are various summation methods, like Able, Borel, and Cesaro. There's also analytic continuation. On Wed, Apr 16, 2014 at 12:10 PM, Bernie Cosell <bernie@fantasyfarm.com> wrote:
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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Blah: Abel, not Able. On Wed, Apr 16, 2014 at 4:12 PM, Mike Stay <metaweta@gmail.com> wrote:
There are various summation methods, like Able, Borel, and Cesaro. There's also analytic continuation.
On Wed, Apr 16, 2014 at 12:10 PM, Bernie Cosell <bernie@fantasyfarm.com> wrote:
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
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Bernie, Abel, Borel, Cesaro, and other methods for assigning a number to a series do not ever make a divergent series convergent. They do assign a number to a series that for convergent series is its sum, so can be thought of as an extension of ordinary convergence. --Dan On Apr 16, 2014, at 3:12 PM, Mike Stay <metaweta@gmail.com> wrote:
There are various summation methods, like Abel, Borel, and Cesaro. There's also analytic continuation.
On Wed, Apr 16, 2014 at 12:10 PM, Bernie Cosell <bernie@fantasyfarm.com> wrote:
. . . 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.
On 16 Apr 2014 at 17:20, Dan Asimov wrote:
Abel, Borel, Cesaro, and other methods for assigning a number to a series do not ever make a divergent series convergent. They do assign a number to a series that for convergent series is its sum, so can be thought of as an extension of ordinary convergence.
Ah, that's crystal clear now. Including the thing that was bothering me: that there was a "special" one -- I find it easy to see that there can be lots of ways to assign some sort of value to a divergent series consistent with the constraint that the method of assignment must give you the actual sum if the series does converge. And I can see that the "assignment method" can be more or less elegant depending on how you're thinking about it [for example: f(series) = sum of series if series converges 4.3 otherwise isn't very elegant...:o)] and one particularly "clean" one [by modern standards] gives you the -1/12 value. I haven't run through the zeta-function definition, I'd guess it has some simple arithment properties, like 2(series) = 2(sum) [and so 2+4+6+8+... would = -1/6, buty I suspect I'd guess wrong..:o) Thanks! /b\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--
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