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I posted this question on math.stackexchange.com and so far the only answer posted (also mentioned in the comments under the question) shows that one of my rash initial guesses about the bottom-line answer was wrong.

Which bijections $f:\{1,2,3,\ldots\}\to\{1,2,3,\ldots\}$ have the property that for every sequence $\{a_n\}_{n=1}^\infty$, $$ \lim_{n\to\infty} \sum_{k=1}^n a_n = \lim_{n\to\infty} \sum_{k=1}^n a_{f(n)}, $$ where "$=$" is construed as meaning that if either limit exists then so does the other and in that case then they are equal?

Here's another rash initial guess, different from the one I posted on stackexchange: It's the bijections whose every orbit is finite.

(That there are uncountably many such bijections can be seen as follows: For each odd number $n$, let $f$ either fix $n$ and $n+1$ or interchange them. That bijection never changes the values of sums. It's a countably infinite sequence of binary choices, so it's uncountable.)

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    $\begingroup$ See Paul Schaefer, Sum-preserving rearrangements of infinite series, Amer Math Monthly 88 (1981) 33-40. $\endgroup$ – Gerry Myerson Aug 4 '15 at 23:16
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    $\begingroup$ Also, Garibay et al., The geometry of sum-preserving permutations, Pac J Math 135 (1988) 313-322, MR0968615 (90f:40001). $\endgroup$ – Gerry Myerson Aug 4 '15 at 23:24
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    $\begingroup$ I also find this: "Rearrangements that Preserve Convergence", Journal of the London Mathematical Society, volume s2-15, issue 1, pages 134-142. jlms.oxfordjournals.org/content/s2-15/1/134.full.pdf+html ${}\qquad{}$ $\endgroup$ – Michael Hardy Aug 5 '15 at 0:47
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    $\begingroup$ Your conjecture cannot be true since the set of permutations with only finite cycles is not closed under multiplication. Consider $g$ = (1 2)(3 4)(5 6)... and $h$ = (2 3)(4 5)(6 7)... . A better conjecture would be to allow all finite products of such permutations. Whether it is the same as the answer of Garibay et al that Gerry mentioned is another question. $\endgroup$ – Brendan McKay Aug 5 '15 at 0:56
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    $\begingroup$ Gerry Myerson already mentioned Schaefer's paper. By the way, as a matter of etiquette, I'd recommend that next time you do all of this literature searching ahead of time, before posting your question to MO, rather than using MO as a scratchpad. $\endgroup$ – Timothy Chow Aug 6 '15 at 2:08
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For the purpose of recording an answer rather than just a pile of links: Michael Hardy requires that

if either limit exists then so does the other and in that case then they are equal?

Let's call this set $G$.

Levi answered a slightly different question, namely characterizing the permutations for which

if the left hand limit exists then so does the right and in that case then they are equal?

I'll call that $P$. Clearly, $G = P \cap P^{-1}$.

Theorem A permutation $f$ is in $P$ if and only if there is a constant $M$ such that, for any $N$, the set $f([1,N])$ can be written as $\bigcup_{i=1}^M [a_i, b_i]$, where $[a,b] = \{ a,a+1, \ldots, b \}$.

Levi provided a different characterization than this one; Agnew provided this characterization; I learned about both from Schaefer who points out that they are fairly directly equivalent.

Pleasants shows that $P$ is not closed under inversion. I haven't (in an one hour skim) found any papers which give a simpler characterization of $P \cap P^{-1}$ than the defining formula.

Remark I would find it nicer to restate Levi/Agnew's characterization as follows: For $S \subseteq \mathbb{N}$, define the blockiness of $S$ to be the least integer $\beta(S)$ such that $S$ can be written as $\bigcup_{i=1}^{\beta(S)} [a_i, b_i]$. (We could have $\beta(S) = \infty$.) Then $f$ is in $P$ if and only if there is a constant $M$ such that $\beta(f(S)) \leq M \beta(S)$. This makes it more obvious that $P$ is closed under composition.

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