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I'm asking the next thing:

If we have a rational conditionally convergent series: $$\beta=\sum_{i=0}^\infty q_i \; \; \; ({q_i}\in\Bbb Q)$$ Then we know thanks to Riemann that it may be rearranged to converge to any value at all, but here's my question: ¿if there's a rearrangement $(q_{\sigma(n)})_{n}$, how can we know that $\sum_{i=0}^\infty q_{\sigma(i)}$ converges to $\beta$? (you can impose conditions on sigma if you want).

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  • $\begingroup$ math.stackexchange.com/questions/446975/… $\endgroup$ Commented Jun 17, 2023 at 18:38
  • $\begingroup$ Thanks, I changed my question for this. $\endgroup$ Commented Jun 17, 2023 at 18:56
  • $\begingroup$ It's a bit unclear what you are looking for. In particular, are you familiar with the proof of Riemann rearrangement theorem? It is argued there how to construct a satisfactory permutation. How does that differ from what you are asking? $\endgroup$ Commented Jun 17, 2023 at 19:00
  • $\begingroup$ @AndrewZhang I'm asking how can we know if a permutation will be a specific number. I know the theorem, but the proof is based only in one permutation. I ask: If we have a permutation, how can we know if this sum will be a number that we want? Sorry for the unclear question. $\endgroup$ Commented Jun 17, 2023 at 19:30

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