<s>[INST] Rearrangements that never change the value of a sum Which bijections $f:\{1,2,3,\ldots\}\to\{1,2,3,\ldots\}$ have the property that for every sequence $\{a_n\}_{n=1}^\infty$,
$$
\lim_{n\to\infty} \sum_{k=1}^n a_k = \lim_{n\to\infty} \sum_{k=1}^n a_{f(k)},
$$
where "$=$" is construed as meaning that if either limit exists then so does the other and in that case then they are equal?
It is clear that there are uncountably many of these.
Might it just be that $\{f(n)/n : n=1,2,3,\ldots\}$ is bounded away from both $0$ and $\infty$?
These bijections form a group.  Can anything of interest be said about them as a group?
PS: Here's another moderately wild guess (the one above appears to be wrong): Might it be just the bijections whose every orbit is finite?
 [/INST] The hypothesis is wrong.
Let $f$ be the following sequence:

$1,2 ; 3,5,4,6; 7,9,11,8,10,12; \cdots$

where each block has two more numbers than the previous block and goes through the odd numbers in order before the even ones.
Clearly $f(n) \in n + O(\sqrt{n})$ as $n \to \infty$.
Now consider the following series:

$\frac{1}{1} - \frac{1}{1} \quad + \quad \frac{1}{2} - \frac{1}{2} + \frac{1}{2} - \frac{1}{2} \quad + \quad \frac{1}{3} - \frac{1}{3} + \frac{1}{3} - \frac{1}{3} + \frac{1}{3} - \frac{1}{3} \quad + \quad \cdots$

split into blocks in the same manner where each block has an alternating sum of the same reciprocal of the block index. Clearly this series converges to $0$. However after applying $f$ to the sequence in the series we get:

$\frac{1}{1} - \frac{1}{1} \quad + \quad \frac{1}{2} + \frac{1}{2} - \frac{1}{2} - \frac{1}{2} \quad + \quad \frac{1}{3} + \frac{1}{3} + \frac{1}{3} - \frac{1}{3} - \frac{1}{3} - \frac{1}{3} \quad + \quad \cdots$

which clearly does not converge.
Also, it is easy to see that we can insert arbitrarily large blocks of identity function between the blocks in the permutation, and corresponding blocks of zeros in the series, and the behaviour is exactly the same. This means that there is a counterexample where $f(n) \in n + o(n)$ as $n \to \infty$, equivalently where $\lim_{n\to\infty} \frac{f(n)}{n}$ exists.
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