---
tag: 6ZDP261K7R
kind: concept
name: Tauberian condition
msc: ["40E05"]
sources:
  - url: https://en.wikipedia.org/wiki/Abelian_and_Tauberian_theorems
---

## Summary

A Tauberian condition is an extra hypothesis on the terms of a series that lets one pass from summability back to ordinary convergence.
[[Abel's theorem|0S4EMDQS2F]] says that a convergent series is Abel summable to its sum, but the converse fails.
[[Tauber's theorem|8SK3GQ4S0M]] shows that the converse holds when $n a_n \to 0$, and this condition is the first example of a Tauberian condition.
Such conditions are the subject of [[Tauberian theory|D6XWK1NRY1]].

## Explanations

### L1 human

A [[summability method|8X25Q4XYMN]] can give a value to a series that does not converge, such as $1 - 1 + 1 - 1 + \cdots$.
So knowing that a series is summable does not tell you that it converges.
A Tauberian condition is a promise that the terms are small or tame enough to rule out such wild cases.
For example, if the terms satisfy $n a_n \to 0$ and the series is Abel summable, then it really converges.
The terms of $1 - 1 + 1 - \cdots$ do not shrink at all, so they fail every such condition.