Rearrangement of a series

Definition

Let $\sum_{n=0}^{\infty} a_n$ be a real series. Let $\sigma \colon \mathbb{N} \to \mathbb{N}$ be a bijection. The series $\sum_{n=0}^{\infty} a_{\sigma(n)}$ is a rearrangement of $\sum a_n$.

Remarks

A rearrangement uses each term of the original series exactly once, in a new order. Rearrangements of an absolutely convergent series have the same sum. For a conditionally convergent series, rearrangements can change the sum, by the Riemann rearrangement theorem.

Explanations

L1 human

A rearrangement adds up the same terms in a different order. For finite sums the order does not matter, but for infinite series it can matter.