TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-07-31

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Tauber's theorem: an Abel-summable series with $ι (n + 1) a_{n} \to 0$ converges ordinarily to its Abel sum

Statement

Let $\sum_{n \geq 0} a_{n}$ be Abel summable to $s$ . If

$ι (n + 1) a_{n} ⟶ 0,$

then its ordinary partial sums converge to $s$ .

Facts & Assumptions

Given: The Abel sum $A (x) : = \sum_{n \geq 0} a_{n} x^{n} \to s$ as $x ↑ 1$ and the stated Tauber condition.

[L1]

The block lemma supplies uniform bounds for the weighted middle and tail when $x_{N} : = 1 - 1 / ι (N + 1)$ (If $ι (n + 1) a_{n} \to 0$ , short multiplicative blocks of the coefficients have uniformly small sums).

[L2]

The Archimedean reciprocal property gives a reciprocal below every positive tolerance. Canonical naturals increase and reciprocation reverses positive order, so every later reciprocal remains below that tolerance; hence $1 / ι (N + 1) \to 0$ and $x_{N} ↑ 1$ . Abel summability then gives $A (x_{N}) \to s$ (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ , Canonical naturals are positive and strictly increasing, Inverses of positives are positive, and reciprocation reverses order, Abel summability by $\lim_{x ↑ 1} \sum a_{n} x^{n}$ and Cesaro summability by the Cesaro means of the partial sums, Limits and Cauchy sequences of reals).

Proof

technique · direct

1.1

Write $S_{N} : = \sum_{n = 0}^{N} a_{n}$ . For $N \geq 1$ , one has $S_{N} - A (x_{N}) = \sum_{n = 0}^{N} a_{n} (1 - x_{N}^{n}) - \sum_{n > N} a_{n} x_{N}^{n}$ .

givenalgebra

2.1

Given $ε > 0$ , choose $N_{0}$ from [L1]. The part of the first sum with $n < N_{0}$ tends to $0$ because it is finite and $x_{N} \to 1$ ; the remaining part and the tail have absolute value at most $ε$ each by [L1].

step 1.1L1choose

3.1

Hence $S_{N} - A (x_{N}) \to 0$ . Since $A (x_{N}) \to s$ by [L2], it follows that $S_{N} \to s$ .

step 2.1L2∎

Depends on

Used by

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Sources

Tauberian theorem, Encyclopedia of Mathematics (standard reference, not scraped)
Tauberian theorems, Encyclopedia of Mathematics (standard reference, not scraped)

Tauber's theorem: an Abel-summable series with ι(n+1)an→0ι(n+1)an​→0 converges ordinarily to its Abel sum