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This question made me think of a related question:

Suppose we have a sequence $a_n$ and a set of permutations $S$ with $|S| = N$ (for some $N > 1$). Suppose that, for any two (distinct) permutations $\sigma$ and $\phi$, both in $S$, we have $$ \sum_{i=1}^\infty a_{\sigma(i)} \ne \sum_{j=1}^\infty a_{\phi(j)} $$ In other words, the rearranged sums of $a_n$ under the permutations in $S$ are all pairwise distinct. In addition, at most one of these rearranged sums diverges.

(NB: The identity function may be a member of $S$, but is not necessarily. WLOG we can always reformulate the problem so that it is.)

If $N = 2$, does this imply $a_n$ is conditionally convergent and can be rearranged to any value? If not, is there some larger value of $N$ which yields this implication, or do we need to have some stronger condition to establish conditional convergence?

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Kevin
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Assuming that there are two rearrangements leading to different sums, that means that the original series is not absolutely convergent (otherwise, any rearrangement would lead to the same value). Any series that is conditionally converging but not absolutely convergent can be rearranged in order to have sum $r$ for any $r\in\mathbb{R}$ by the Riemann(-Dini) theorem. For short, if we have two rearrangements leading to different sums, we have an infinite number (the cardinality of $2^\mathbb{N}$, i.e. a continuum) of rearrangements leading to different sums.

Jack D'Aurizio
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  • It was my impression that, among other things, conditional convergence requires that $\sum |a_n|$ diverges... do we have that here? – Kevin Jun 25 '16 at 22:45
  • @Kevin: a series is said conditionally convergent when the limit $$\lim_{N\to +\infty}\sum_{n=1}^{N}a_n$$ exists, nothing is required for the series of absolute values. – Jack D'Aurizio Jun 25 '16 at 22:48
  • I thought that was called "convergent." Then it's absolutely convergent if the series of absolute values converges and conditionally convergent otherwise. – Kevin Jun 25 '16 at 22:50
  • Probably a (local) terminology issue. To me, convergent is a synonym of conditionally convergent and absolutely convergent means that the series of absolute values is convergent. Anyway, the truth of the above lines should not be affected. – Jack D'Aurizio Jun 25 '16 at 23:00