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.)
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 Helms, Kassel
