Math 2.0

Sequences, series, summability Abelian and Tauberian theorems Theorem

Littlewood's Tauberian theorem

Builds on Abel summationAbelian and Tauberian theoremsBig O notationConvergent series

Statement

Let (an)n≥0(a_n)_{n \ge 0} be complex numbers such that ∑n≥0anxn\sum_{n \ge 0} a_n x^n converges for 0≤x<10 \le x < 1. Assume that the series ∑an\sum a_n is Abel summable to ss, that is, ∑n≥0anxn→s\sum_{n \ge 0} a_n x^n \to s as x→1−x \to 1^-. Assume also that an=O(1/n)a_n = O(1/n) in big O notation. Then ∑n≥0an\sum_{n \ge 0} a_n converges and its sum is ss.

Remarks

Littlewood's original proof was long. Karamata found a short proof. It first controls the sums ∑ang(xn)\sum a_n g(x^n) for polynomials gg and then approximates the indicator function of an interval from above and below by polynomials, which isolates the partial sum sNs_N. The same method proves the Hardy–Littlewood Tauberian theorem.

The Tauberian condition cannot be dropped. For an=(−1)na_n = (-1)^n, ∑anxn=11+x→12\sum a_n x^n = \frac{1}{1+x} \to \frac12, but ∑(−1)n\sum (-1)^n diverges; here nann a_n is unbounded.

Going deeper

For a student

The theorem strengthens Tauber's theorem, where the condition is nan→0n a_n \to 0, and it is the Abel analogue of Hardy's Tauberian theorem for Cesàro means. Because Abel summation is stronger than Cesàro summation, it implies Hardy's theorem. Littlewood's theorem was a starting point of the systematic study of Tauberian theorems by Hardy and Littlewood.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Abel summation, concept, no evidence levelAbel summationconceptAbelian and Tauberian theorems, concept, no evidence levelAbelian and Tauberian…conceptBig O notation, definition, no evidence levelBig O notationdefinitionSequence, definition, no evidence levelSequencedefinitionReal 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 seriesdefinitionLittlewood's Tauberian theorem, statement, no evidence levelLittlewood's Tauberia…statement
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. Abel summation concept
  2. Abelian and Tauberian theorems concept
  3. Big O notation definition
  4. Sequence definition
  5. Real sequence definition
  6. Limit of a sequence definition
  7. Series definition
  8. Convergent sequence definition
  9. Partial sum definition
  10. Convergent series definition
  11. Littlewood's Tauberian theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Littlewood%27s_Tauberian_theorem
About this page
TagZ35TD4ZZRD
URIm2://Z35TD4ZZRD
Kindstatement, theorem
WritingW0 stub
MSC40E05, 40G10
StatusThis copy of the site has no build data. Run m2 status Z35TD4ZZRD.
AboutAbel summation, Abelian and Tauberian theorems
UsesAbel summation, Abelian and Tauberian theorems, Big O notation, Convergent series
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Changed2 November 2026

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