---
tag: 2ME9M61TJK
kind: concept
name: Unconditional convergence
msc: [40A05, 46B15]
specializes: [0AF7X08PCR]
sources:
  - url: https://en.wikipedia.org/wiki/Unconditional_convergence
same_as:
  wikidata: Q1630320
  wikipedia: Unconditional convergence
---

## Summary

A series converges unconditionally when every reordering of its terms still converges.
For real or complex series this is the same as [[absolute convergence|0YBDSJARWN]], by the [[Riemann rearrangement theorem|0E7VBYGFF7]].
In infinite-dimensional Banach spaces the two notions differ, and unconditional convergence is the natural notion for sums without a preferred order.
The exact definition is in [[unconditional convergence|GA7Q813F50]].

## Explanations

### L1 human

Some infinite sums give different answers when you add the terms in a different order.
An unconditionally convergent series is one where the order never matters: every order converges, to the same sum.
For example $1 + \tfrac14 + \tfrac19 + \cdots$ converges in any order, while $1 - \tfrac12 + \tfrac13 - \cdots$ can be reordered to converge to any real number.