Rearrangement theorem for absolutely convergent series

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

Rearrangement theorem for absolutely convergent series

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

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

converges, and

\sum_{n=1}^\infty a_{\pi(n)}=\sum_{n=1}^\infty a_n.

In contrast, for a conditionally convergent series the value can depend on the rearrangement; this is quantified by the Riemann rearrangement theorem .