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I recently posted this question. It was closed shortly after posting for being opinion based. I tried to edit the question to make it clear exactly what I was asking, and the question was downvoted shortly afterwards. Did I make a mistake somewhere in my edit? I think that the recursive definition I provided addresses the issues mentioned in comments; the order and grouping of summands are unambiguous. The "set of possible values" referred to should be the image of the set of well-ordered partitionings of $\Bbb{Z}$. Am I missing something? What can I do to make my question more clear?

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    $\begingroup$ I suspect people are downvoting the question because, superficially, it makes no sense. Therefore, your question might benefit from a few words at the start really emphasising that you know that this sum is undefined. [Possibly it might help to state your level, e.g. "I've just come across divergent series..." or "I'm a grad student and came across this idea when teaching divergent series...". Don't say that you have no formal training though, as you'll get more downvotes for crankery. Because people are cranky.] $\endgroup$ – user1729 Dec 13 '19 at 19:17
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    $\begingroup$ The question has been reopened. Maybe a clearer way to phrase it would be, for which numbers $k$ is it possible to partition the integers into finite sets, such that the integers in one of those sets sum to $k$, while in each of the other sets the integers sum to zero? $\endgroup$ – Gerry Myerson Dec 13 '19 at 22:51

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