Real series/Reordering/Definition

Reordering of a series

Let $\sum _{k=0}^{\mathrm{\infty }}{a}_k$ be a real series. A reordering of this series is a series $\sum _{k=0}^{\mathrm{\infty }}{b}_k$ with $b_k=a_{\sigma (k)}$ for a bijective mapping $\sigma :\mathbb{N}\to \mathbb{N}$.