Rearrangement of a series

Let $\sum_{n=1}^\infty a_n$ be a series in $\mathbb{R}$ or $\mathbb{C}$. A rearrangement of the series is any series of the form

where $\sigma:\mathbb{N}\to\mathbb{N}$ is a bijection (a permutation of the indices).

Rearrangements preserve sums for absolutely convergent series, but in $\mathbb{R}$ conditionally convergent series can have rearrangements with different sums or even divergence.

Examples:

• If $\sigma(n)=n$ for all $n$, the rearrangement is the original series.
• For $a_n=(-1)^{n+1}\frac{1}{n}$ (alternating harmonic), reordering terms can change the sum.
• Grouping terms is not the same as rearrangement unless it corresponds to a permutation of indices without changing term values.