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 <--