The standard rearrangement theorem for conditionally convergent series says that the terms in a conditionally convergent series can be rearranged so that the new sum is any desired number, or $\pm\infty$.


Let me set up some notation to investigate this a little further. My coefficients will be determined by a function $\alpha:\mathbb{N}\to\mathbb{R}$ (or $\mathbb{C}$). Then I'll simply 
write $\sum\alpha$ for this series $\sum\_{n=1}^\infty \alpha(n)$.


Now we let $\operatorname{Sym}(\mathbb{N})$ act on the set $A = \{ \alpha: \mathbb{N}\to \mathbb{R} \}$ by the rule
$\alpha^\sigma (x) = \alpha(\sigma(x))$. Then we can ask simple questions


1. is $\sum \alpha^\sigma = \sum \alpha$?
2. if $\sum\alpha$ converges, does $\sum\alpha^\sigma$ also converge?


But these are not really the questions I want to ask. 


Let $B\subseteq A$ be the subset of all sequences $\alpha$ so that the series $\sum\alpha$ is conditionally convergent. 


**Question 1** What is the stabilizer $G = \{ \sigma \mid \alpha^\sigma \in B\ \forall \alpha\in B\}$?


**Question 2** What is the group of sum-preserving permutations $H = \{ \sigma \mid \sum \alpha^\sigma = \sum\alpha \ \forall \alpha\in B\}$?


Certainly any permutation which is the identity off a finite subset 
of $\mathbb{N}$ is contained in both. 


There are some subgroups of $\operatorname{Sym}(\mathbb{N})$ 


This is related to the question [here](https://mathoverflow.net/questions/182900/partitions-of-rm-sym-mathbbn-induced-by-convergent-but-not-absolutely).



The stabilizer of the conditionally convergent series
[Making comment into answer.] If there is a constant $C$ such that $|\sigma(n)−n| \leq C$ for all $n$, then $\sigma$ is sum-preserving. And this "boundedness" property is preserved by compositions and inverses. So one might guess that perhaps these are all of the sum-preserving permutations. But it is not the case. There are sum-preserving permutations that have "unbounded steps" (but, I suppose, very far apart). There are several known characterizations of sum-preserving permutations, see for example <http://projecteuclid.org/euclid.pjm/1102688295> and references therein.


