---
tag: 29C7C41A52
kind: concept
name: Regular summability method
msc: [40C99, 40D05]
specializes: [8X25Q4XYMN]
sources:
  - url: https://en.wikipedia.org/wiki/Divergent_series
---

## Summary

A [[summability method|8X25Q4XYMN]] is regular when it sums every convergent sequence to its ordinary limit.
Regularity is the basic requirement for a method to count as a generalization of convergence.
For matrix methods it is characterized by the [[Silverman–Toeplitz theorem|S4J7SRPXMZ]].

## Explanations

### L1 human

A new way of summing series should not change the answer for series that already have a sum.
Regular methods have this property: on convergent series they agree with the ordinary sum, and they only add new values for some divergent series.
Cesàro summation is regular, because the running averages of a convergent sequence tend to the same limit.