---
tag: A82GRSZMVW
kind: concept
name: Absolute summability
msc: [40F05, 40C05]
sources:
  - url: https://en.wikipedia.org/wiki/Absolute_convergence
---

## Summary

Absolute summability is a strengthening of summability by a [[matrix summability method|M6311DWJ1E]].
A series is absolutely summable by a method when the transformed sequence not only converges but has bounded variation.
It plays for summability methods the role that [[absolute convergence|0YBDSJARWN]] plays for ordinary convergence.

## Explanations

### L1 human

A summability method turns the partial sums of a series into a new sequence, for example their running averages.
Ordinary summability asks that this new sequence settles down to a limit.
Absolute summability asks more: the total of all the jumps between consecutive terms of the new sequence must be finite.
For the identity method, where the new sequence is the sequence of partial sums itself, the jumps are the terms $a_n$, and one gets back absolute convergence.