If an infinite sum is absolutely convergent, its limit remains the same however the terms are permuted. Does a sequence of real numbers $(a_n)_{n\in\mathbb{N}}$ exist such that $\sum_n a_{\small P(n)}$ is the same for all permutations $P:\mathbb{N}\overset{\text{bijectively}}{\to}\mathbb{N}$, and such that $\sum_n |a_n|$ diverge?
2026-03-25 10:55:35.1774436135
Are absolutely convergent sums exactly those which can be reordered?
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Assuming that when you write “is the same” what you mean is “is the same real number”, then the answer is negative, by the Riemann rearrangement theorem.