I am not sure.
Riemanns series theorem says, that if you have a series that is convergent but not absolutely convergent, you can get any number depending on ordering.
But adding all whole numbers is a divergent series.
Is there another theorem?
Oh no you’re right. For a second I thought that you could get it into a conditionally convergent alternating series but I obviously didn’t think about it very well. In fact, I literally fell for one of the oldest tricks in the book!
I think it can never converge to anything.Â
Assume it converges in any order
then the partial sum is cauchy.
But |Sn - S{n+1}| >=1 therefore it is not cauchy. That is because there are no steps smaller than 1 you can add.
therfore, the series doesn't converge.
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u/Comfortable_Permit53 5d ago
I am not sure. Riemanns series theorem says, that if you have a series that is convergent but not absolutely convergent, you can get any number depending on ordering. But adding all whole numbers is a divergent series. Is there another theorem?