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Is the concept of conditional convergence of series whose terms are real numbers a topic of research in analysis or merely something to be aware of?

The question is prompted by the conjunction of my participation in some mathoverflow threads about rearrangements of series and the fact that I've started writing something about how a simple trigonometry problem led to a question whose answer is independent of the usual axioms of set theory.

I find myself thinking of things like this: For each conditionally convergent series there is a set of permutations of the indices that do not alter the sum, and there is a function from the set of rearrangements to the set of sums, and for each permutation of the indices there is a set of series whose sums it does not alter, and for each such permutation there is a mapping from sums to changes worked by that permutation, and there could be theorems about these objects. BUT is there some point to doing any of this?

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    $\begingroup$ This suggests to me Galois connections (arguably purely algebraic objects) used in functional analysis. It may lend a new perspective to old topics. Since this places it far from my trajectory of study, I can only say that you will know if there is a point to it only after you've done it. Gerhard "Topos Theory Isn't Pointless Research" Paseman, 2017.06.20. $\endgroup$ – Gerhard Paseman Jun 20 '17 at 21:40
  • $\begingroup$ "Is there some point to doing any of this" is a question that I ask myself often, and so do reviewers of my grant proposals. ... Okay, less facetiously: What kind of point do you WANT it to have, relative to how much time and energy you can afford to invest? Will you settle for nothing less than an influential publication in a top journal? A modest paper in a specialized journal? An article for students about how a trigonometry problem led in an unexpected direction? Material for student projects (undergraduate senior theses, masters theses, etc.)? (NOT necessarily monotone in difficulty.) $\endgroup$ – Zach Teitler Jun 21 '17 at 3:15
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    $\begingroup$ @ZachTeitler : The particular points you mention are all devoted to the worship of the present incumbent gods, and certainly there is much to be said for that. However, those particular deities may get old and tired before we do..... I have come to think that the current market price of purity in mathematics may have risen to unsustainably high levels and a bubble may burst at some point. If we could connect this to something like engineering or genetics or big data we'd have a better assurance that it could survive the market crash. $\endgroup$ – Michael Hardy Jun 21 '17 at 3:42
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    $\begingroup$ One of Borel's first two or three papers, Sur le changement de l'ordre des termes d'une série semi-convergente, which was written when he was a first-year college undergraduate, deals with this general topic, and my impression is that the literature on it is enormously vast, especially when various summability methods and series in Banach spaces are included in the mix. If you have access to Math. Reviews, I suspect very little time would be needed to track down what has been published in the last couple of decades on this topic. $\endgroup$ – Dave L Renfro Jun 30 '17 at 14:28
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    $\begingroup$ In case you're interested, I just posted an answer to Questions about conditionally convergent series and rearrangement of that gives a lot more details related to my 30 June 2017 comment. $\endgroup$ – Dave L Renfro Dec 9 '18 at 10:33

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