---
tag: 70HJTK5VZH
kind: concept
name: Summation of families over arbitrary index sets
msc: [40A05, 46B15]
sources:
  - url: https://en.wikipedia.org/wiki/Series_(mathematics)
---

## Summary

Let $(x_i)_{i \in I}$ be a family indexed by an arbitrary set $I$.
It is summable when its finite partial sums approach a limit as the finite set of indices grows.
This notion of sum uses no order on $I$, so it works for index sets such as $\N \times \N$, $\Z$ or uncountable sets.
For real or complex terms indexed by $\N$, summability is the same as [[absolute convergence|0YBDSJARWN]].
The exact definition is in [[summable family|Z4H7272S9S]].

## Explanations

### L1 human

An ordinary series adds its terms in a fixed order.
A summable family has a sum that does not depend on any order.
Once a large enough finite batch of terms is added, adding further terms keeps you close to the sum.
For example, the numbers $1/(2^m 3^n)$ for all pairs $(m, n)$ of natural numbers form a summable family with sum $3$.