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Rearrangement theorem for absolutely convergent series

Any rearrangement of an absolutely convergent series converges to the same sum.

Rearrangement theorem (absolute convergence): Let ∑n=1∞an\sum_{n=1}^\infty a_n be an of real or , and let π:N→N\pi:\mathbb{N}\to\mathbb{N} be a bijection. Then the rearranged series

∑n=1∞aπ(n)\sum_{n=1}^\infty a_{\pi(n)}

converges, and

∑n=1∞aπ(n)=∑n=1∞an.\sum_{n=1}^\infty a_{\pi(n)}=\sum_{n=1}^\infty a_n.
Remarks

In contrast, for a the value can depend on the rearrangement; this is quantified by the .