Math 2.0

Sequences, series, summability Definition

Abelian and Tauberian theorems

Builds on Summability methodTauberian conditionConvergent seriesSeries

Definition

Let MM be a summability method that assigns values to some series of complex numbers.

  1. An Abelian theorem for MM is a theorem stating that every convergent series is MM-summable to its sum. More generally, an Abelian theorem for two methods MM and M′M' states that every MM-summable series is M′M'-summable to the same value.
  2. A Tauberian theorem for MM is a theorem stating that an MM-summable series converges if its terms satisfy a stated additional condition. Its sum is then its MM-value. The additional condition is called a Tauberian condition.

The same terms are used for methods that act on sequences, functions or measures, with "convergent" replaced by the corresponding ordinary limit.

Remarks

Example of an Abelian theorem: Abel's theorem on power series says that if ∑an\sum a_n converges to ss, then ∑anxn→s\sum a_n x^n \to s as x→1−x \to 1^-. Example of a Tauberian theorem: Tauber's theorem says that if ∑anxn→s\sum a_n x^n \to s as x→1−x \to 1^- and nan→0n a_n \to 0, then ∑an\sum a_n converges to ss.

Non-example: the statement "if ∑anxn→s\sum a_n x^n \to s as x→1−x \to 1^-, then ∑an\sum a_n converges" has no Tauberian condition, and it is false. For an=(−1)na_n = (-1)^n one has ∑anxn=11+x→12\sum a_n x^n = \frac{1}{1+x} \to \frac12, but ∑(−1)n\sum (-1)^n diverges.

The names honour Abel and Tauber.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Sequence, definition, no evidence levelSequencedefinitionSummability method, concept, no evidence levelSummability methodconceptTauberian condition, concept, no evidence levelTauberian conditionconceptReal sequence, definition, no evidence levelReal sequencedefinitionLimit 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 sumdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionAbelian and Tauberian theorems, definition, no evidence levelAbelian and Tauberian…definition
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. Sequence definition
  2. Summability method concept
  3. Tauberian condition concept
  4. Real sequence definition
  5. Limit of a sequence definition
  6. Series definition
  7. Convergent sequence definition
  8. Partial sum definition
  9. Convergent series definition
  10. Abelian and Tauberian theorems definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Abelian_and_Tauberian_theorems
About this page
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Kinddefinition
WritingW0 stub
MSC40E05, 40D05, 11M45
DefinesAbelian and Tauberian theorems
UsesSummability method, Tauberian condition, Convergent series, Series
Changed2 November 2026

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