---
tag: TXS2H001R4
kind: concept
name: Summation in topological groups
msc: [40J05, 22A05]
sources:
  - url: https://en.wikipedia.org/wiki/Series_(mathematics)
---

## Summary

Let $(x_i)_{i \in I}$ be a family in a commutative topological group, indexed by any set $I$.
It has sum $s$ when the finite partial sums $\sum_{i \in F} x_i$ converge to $s$ as the finite set $F \subseteq I$ grows.
This defines infinite sums with no order on the index set, and the sum does not depend on any rearrangement.
In a Hausdorff group a family has at most one sum.
This is the notion of summability used in Mathlib (`HasSum`), and it covers series in $\R$, in Banach spaces and in $p$-adic numbers.

## Explanations

### L1 human

To add infinitely many elements of a group without choosing an order, look at all finite sub-collections and their sums.
The family has sum $s$ if these finite sums get as close to $s$ as you like once the sub-collection contains a large enough finite set.
For real numbers this is stronger than ordinary convergence of a series.
For example, $1 - \tfrac12 + \tfrac13 - \cdots$ converges as a series, but as a family it has no sum, because one can pick finite sets of only positive terms with arbitrarily large sums.