02/27/2023, 05:29 AM
(02/27/2023, 04:22 AM)Gottfried Wrote: 1) I like vrey much the attempt of Caleb to find contoures of a "greater picture" of this type of divergence and the hope of finding a way to extend the function(s) beyond its/their natural boundary.
2) I didn't see in the above discussin (overlooked?) links to two Q&A in MO which Caleb and James surely know, but most other tetforum-members possibly not. The following links seem to me much fruitful for more analytical and heuristic properties of this style of functions and their evaluations as sums/series:
- https://mathoverflow.net/q/198665/7710 Properties of summation of an eventually oscillating series. A couple of interesting answers about exactly Caleb's series can be found, I liked (and like) always the contribution(s) of Robert Israel as very often much enlightening.
- https://mathoverflow.net/a/198871/7710 my answer to the above, showing some specific analysis/heuristics using the reordering of the double-series like Caleb has shown it here.
- https://mathoverflow.net/q/201098/7710 a follow up question of mine, trying to put that specific series *in a greater context* of what I called here sometimes "(alternating) iteration series [AIS]" on which I had the similar property of oscillation, but having no answer so far.
(Unfortunately, after pandemy years and own general health/age issues I can no more undertake reasonable attacks on this matter myself, so I likely cannot be of much help in this issue.)
Gottfried! Its great to see you here! I'll surely check out your MO quesiton, it looks to have some deep and interesting content.
Also, as to your second link, the similarity between my approach and your is not a coincidence. The methods I devoloped were inspired by seeing your answer! In fact, your last sentence on your answer was
Quote:Possibly we have here something like in the Ramanujan-summation of divergent series where we have to add some integral to complete the divergent sums, but I really don't know.And, this is essentially exactly what I have done in this post (the main difference being that the integral in this case goes in the complex plane as opposed to the Ramanjuan integral which is on the real line).
