Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Continuum sums -- a big problem and some interesting observations
(10/10/2010, 08:02 AM)Ansus Wrote: So my advice is to avoid confusion by using continous sum but not indefinite continous sum, but a continous sum with fixed lower limit which is defined unambiguiously.

So, then we have

as definitely ambiguous and not well defined if is not entire, even with Fourier sums, as powerful as they are, while

is not. The direction of the singularities is toward the right of a singularity in if , where is the singularity point, and to the left otherwise. And if , it goes in both directions, i.e. .

Thus, the laws



where for some , are only guaranteed to hold when all numbers , , and are integers, or is entire.

Also, I made an interesting observation about the function


a sort of sum-analogue of the error function. With the Fourier-based sum, it does not settle to a limit as , but instead goes to a small 1-cyclic wobble about the limit of the discrete sum. Namely, , and so


The value 1/2 for the function at 1/2 can also be obtained from Faulhaber's formula when it is organized as a sum over the Bernoulli polynomials. I think the half-integers are the only other places than the integers where that formula converges.

The message here being, of course, that we can't necessarily expect continuum sums and discrete sums to behave in the same way, so the idea we may have to sacrifice some identities may not be a surprise.

Messages In This Thread
RE: Continuum sums -- a big problem and some interesting observations - by mike3 - 10/11/2010, 10:36 AM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Interesting commutative hyperoperators ? tommy1729 0 95 02/17/2020, 11:07 PM
Last Post: tommy1729
  Very interesting topic Ansus 0 466 10/01/2019, 08:14 PM
Last Post: Ansus
  THE problem with dynamics tommy1729 1 2,493 04/04/2017, 10:52 PM
Last Post: tommy1729
  [sci.math] halting problem skeptic. tommy1729 0 1,787 05/30/2015, 09:24 PM
Last Post: tommy1729
  2015 Continuum sum conjecture tommy1729 3 3,565 05/26/2015, 12:24 PM
Last Post: tommy1729
  Problem with cauchy method ? tommy1729 0 1,977 02/16/2015, 01:51 AM
Last Post: tommy1729
  Another way to continuum sum! JmsNxn 6 6,807 06/06/2014, 05:09 PM
Last Post: MphLee
  Further observations on fractional calc solution to tetration JmsNxn 13 14,562 06/05/2014, 08:54 PM
Last Post: tommy1729
  Remark on Gottfried's "problem with an infinite product" power tower variation tommy1729 4 5,419 05/06/2014, 09:47 PM
Last Post: tommy1729
  On naturally occuring sums tommy1729 0 2,028 10/24/2013, 12:27 PM
Last Post: tommy1729

Users browsing this thread: 1 Guest(s)