mathematics

Littlewood's Tauberian theorem

Littlewood's Tauberian theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Littlewood's Tauberian theorem rather than just read about it. In short: In mathematics, Littlewood's Tauberian theorem is a strengthening of Tauber's theorem introduced by John Edensor Littlewood (1911). Statement Littlewood showed the following: If an = O(1/n ), and as x ↑ 1 we have ∑ a n x n → s , {\displaystyle \sum a_{n}x^{n}\to s,} then ∑ a n = s . {\displaystyle \sum a_{n}=s.} Hardy and Littlewood later showed that the hypothesis on an could be weakened to the "one-sided" conditio…

Key takeaways

Littlewood's Tauberian theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Littlewood's Tauberian theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Littlewood's Tauberian theorem from memory before moving on to harder problems.

Reference excerpt

In mathematics, Littlewood's Tauberian theorem is a strengthening of Tauber's theorem introduced by John Edensor Littlewood (1911).

Statement Littlewood showed the following: If an = O(1/n ), and as x ↑ 1 we have

∑ a n x n → s , {\displaystyle \sum a_{n}x^{n}\to s,}

then

∑ a n = s . {\displaystyle \sum a_{n}=s.}

Hardy and Littlewood later showed that the hypothesis on an could be weakened to the "one-sided" condition an ≥ –C/n for some constant C. However in some sense the condition is optimal: Littlewood showed that if cn is any unbounded sequence then there is a series with |an| ≤ |cn|/n which is divergent but Abel summable.

History Littlewood (1953) described his discovery of the proof of his Tauberian theorem. Alfred Tauber's original theorem was similar to Littlewood's, but with the stronger hypothesis that an=o(1/n). Hardy had proved a similar theorem for Cesàro summation with the weaker hypothesis an=O(1/n), and suggested to Littlewood that the same weaker hypothesis might also be enough for Tauber's theorem. In spite of the fact that the hypothesis in Littlewood's theorem seems only slightly weaker than the hypothesis in Tauber's theorem, Littlewood's proof was far harder than Tauber's, though Jovan Karamata later found an easier proof. Littlewood's theorem follows from the later Hardy–Littlewood Tauberian theorem, which is in turn a special case of Wiener's Tauberian theorem, which itself is a special case of various abstract Tauberian theorems about Banach algebras.

Examples

References Korevaar, Jacob (2004), Tauberian theory. A century of developments, Grundlehren der Mathematischen Wissenschaften, vol. 329, Springer-Verlag, doi:10.1007/978-3-662-10225-1, ISBN 978-3-540-21058-0 Littlewood, J. E. (1953), "A mathematical education", A mathematician's miscellany, London: Methuen, MR 0872858 Littlewood, J. E. (1911), "The converse of Abel's theorem on power series" (PDF), Proceedings of the London Mathematical Society, 9 (1): 434–448, doi:10.1112/plms/s2-9.1.434

Worked examples

Example 1 — a first encounter with Littlewood's Tauberian theorem

Start with the simplest possible case. Write down what Littlewood's Tauberian theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Littlewood's Tauberian theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Littlewood's Tauberian theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Littlewood's Tauberian theorem

In research: Littlewood's Tauberian theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Littlewood's Tauberian theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Littlewood's Tauberian theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Tauberian theorems, so understanding it makes those chapters shorter.
In everyday life: Look for Littlewood's Tauberian theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Littlewood's Tauberian theorem” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Littlewood's Tauberian theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Littlewood's Tauberian theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Littlewood's Tauberian theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Littlewood's Tauberian theorem in simple terms?

In mathematics, Littlewood's Tauberian theorem is a strengthening of Tauber's theorem introduced by John Edensor Littlewood (1911). Statement Littlewood showed the following: If an = O(1/n ), and as x ↑ 1 we have ∑ a n x n → s , {\displaystyle \sum a_{n}x^{n}\to s,} then ∑ a n = s . {\displaystyle…

Why does Littlewood's Tauberian theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Littlewood's Tauberian theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Littlewood's Tauberian theorem.