Abel summation
E451515
Abel summation is a method in mathematical analysis for assigning values to certain divergent series by considering the limit of their power series as the variable approaches 1 from below.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Abel summation canonical | 1 |
| Abelian summation | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4552232 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Abel summation Context triple: [Divergent Series, topic, Abel summation]
-
A.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
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B.
Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
-
C.
Euler’s method of rearranging absolutely convergent series
Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
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D.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
E.
Poisson summation formula
The Poisson summation formula is a fundamental result in harmonic analysis that links sums of a function over the integers to sums of its Fourier transform, with deep applications in number theory, signal processing, and physics.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Abel summation Target entity description: Abel summation is a method in mathematical analysis for assigning values to certain divergent series by considering the limit of their power series as the variable approaches 1 from below.
-
A.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
B.
Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
-
C.
Euler’s method of rearranging absolutely convergent series
Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
-
D.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
E.
Poisson summation formula
The Poisson summation formula is a fundamental result in harmonic analysis that links sums of a function over the integers to sums of its Fourier transform, with deep applications in number theory, signal processing, and physics.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
concept in mathematical analysis
ⓘ
summability method ⓘ summation method ⓘ |
| appliesTo |
conditionally convergent series
ⓘ
divergent series ⓘ infinite series ⓘ |
| basedOn |
limit of power series as variable approaches 1 from below
ⓘ
power series ⓘ |
| category |
infinite series techniques
ⓘ
methods of summability ⓘ |
| characterizedBy | taking limit of sum a_n x^n as x approaches 1 from below ⓘ |
| contrastsWith | ordinary termwise convergence of series ⓘ |
| field |
mathematical analysis
ⓘ
series summation ⓘ summability theory ⓘ |
| formalDefinition | A series ∑ a_n is Abel summable to s if lim_{x→1−} ∑ a_n x^n = s ⓘ |
| generalizes | ordinary sum of absolutely convergent series ⓘ |
| hasAlternativeName |
Abelian summation
ⓘ
linked to:
Abel summation
|
| hasProperty |
extends ordinary convergence of series
ⓘ
linear summation method ⓘ regular summation method ⓘ stable under scalar multiplication ⓘ stable under termwise addition ⓘ |
| implies | ordinary convergence when Abel sum exists and series is convergent ⓘ |
| introducedIn | 19th century ⓘ |
| namedAfter | Niels Henrik Abel ⓘ |
| relatedConcept |
Abel transform
ⓘ
Abel’s limit theorem ⓘ
linked to:
Abel’s theorem
|
| relatedTo |
Abel’s theorem
ⓘ
Borel summation ⓘ Cesàro summation ⓘ Tauberian theorems ⓘ power series convergence ⓘ |
| requires | existence of limit of associated power series at x = 1− ⓘ |
| strongerThan | Cesàro summation of order 1 in many contexts ⓘ |
| toolFor |
defining sums via analytic continuation of power series
ⓘ
regularization of divergent series ⓘ |
| usedFor |
assigning values to divergent series
ⓘ
studying convergence of series ⓘ |
| usedIn |
Fourier series theory
ⓘ
linked to:
Fourier series
analytic number theory ⓘ asymptotic analysis ⓘ study of generating functions ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Abel summation Description of subject: Abel summation is a method in mathematical analysis for assigning values to certain divergent series by considering the limit of their power series as the variable approaches 1 from below.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Abel summation