---
tag: 0HRJ2Y74T3
kind: definition
name: Rearrangement of a series
msc: [40A05]
defines: 0W1K48V5SG
uses: [0JH4MN4K1C]
sources:
  - url: https://en.wikipedia.org/wiki/Riemann_series_theorem
    locator: Definitions
---

## Definition

Let $\sum_{n=0}^{\infty} a_n$ be a real [[series|0JH4MN4K1C]].
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|0J76TP9FGD]].
For a [[conditionally convergent|02YRQXZDCS]] series, rearrangements can change the sum, by the [[Riemann rearrangement theorem|0E7VBYGFF7]].

## 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.