Are absolutely convergent sums exactly those which can be reordered?

Question

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?

Answer 1

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.