Math 2.0

Sequences, series, summability Definition

Tauberian condition

Builds on Summability methodSeriesConvergent seriesAbel's theoremTauber's theorem

Definition

Let MM be a summability method that assigns a value to some series ∑an\sum a_n of real or complex numbers. Let PP be a property of sequences (an)n≥0(a_n)_{n \ge 0}. The property PP is a Tauberian condition for MM if the following holds. Every series ∑an\sum a_n whose terms have property PP and which is MM-summable to ss converges with sum ss.

Remarks

Example: for Abel summation, the property nan→0n a_n \to 0 is a Tauberian condition. This is Tauber's theorem. Stronger theorems allow weaker conditions; for example, nan=O(1)n a_n = O(1) is a Tauberian condition both for Cesàro summation and for Abel summation.

Non-example: the property an→0a_n \to 0 is not a Tauberian condition for Cesàro summation. Let the partial sums start at −1-1. For k=1,2,3,…k = 1, 2, 3, \dots in turn, let them run up to 11 in kk steps of size 2/k2/k and then back down to −1-1 in kk such steps. The terms are ±2/k\pm 2/k, so an→0a_n \to 0. Over each full up-and-down run the partial sums add up to 00. The incomplete last run contributes at most 2k+22k + 2 in absolute value, while nn is at least of order k2k^2. So the arithmetic means of the partial sums tend to 00, but the partial sums reach 11 and −1-1 infinitely often and do not converge.

A theorem whose hypothesis is a Tauberian condition is called a Tauberian theorem. It is a partial converse of an Abelian theorem such as Abel's theorem.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

Show the path as a graph
Big O notation, definition, no evidence levelBig O notationdefinitionLimit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionSummability method, concept, no evidence levelSummability methodconceptLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionOne-sided limit, definition, no evidence levelOne-sided limitdefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionPower series, definition, no evidence levelPower seriesdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionAbel's theorem, statement, no evidence levelAbel's theoremstatementTauber's theorem, statement, no evidence levelTauber's theoremstatementTauberian condition, definition, no evidence levelTauberian conditiondefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Big O notation definition
  2. Limit point of a set definition
  3. Sequence definition
  4. Summability method concept
  5. Limit of a function definition
  6. Real sequence definition
  7. One-sided limit definition
  8. Limit of a sequence definition
  9. Series definition
  10. Convergent sequence definition
  11. Partial sum definition
  12. Power series definition
  13. Convergent series definition
  14. Abel's theorem statement
  15. Tauber's theorem statement
  16. Tauberian condition definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Abelian_and_Tauberian_theorems
About this page
TagC28YT5DQFF
URIm2://C28YT5DQFF
Kinddefinition
WritingW0 stub
MSC40E05
DefinesTauberian condition
UsesSummability method, Series, Convergent series, Abel's theorem, Tauber's theorem
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.