I guess that's my question. There are clearly some no-no's with formal manipulation of infinite series, for instance, if a series includes an infinite number of both positive and negative terms, you can't arbitrarily reorder the elements . However, there are likely some "safe" manipulations, like termwise sum and product, scalar multiplication, zero term elision, &c, that do not alter the value of the sum. The question is, if we start with Cesaro-summable sequences and apply only safe manipulations, do we end up with unique values for sequences like 1 + 2 + 3 + 4 + ...?
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of Cris Moore Sent: Saturday, March 01, 2014 5:03 PM To: math-fun Subject: Re: [math-fun] numberphile.com
I don't think the use of zeta(-1) = -1/12 in physics has anything to do with these formal manipulations --- they have to do with extending zeta(s) analytically in the complex plane.
What's curious is that there is a set of formal manipulations that leads to the same answer. But this may be true for _every_ possible answer.
Cris
On Mar 1, 2014, at 2:07 PM, David Wilson <davidwwilson@comcast.net> wrote:
Seriously, do physicists routinely use these types of formal manipulations without any mathematical foundation? I'm guessing not. If physicists are using equation like [1] in actual physics, I suspect they must have some theory to justify their methods.
Cristopher Moore Professor, Santa Fe Institute
The Nature of Computation Cristopher Moore and Stephan Mertens Available now at all good bookstores, or through Oxford University Press http://www.nature-of-computation.org/
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