theory of divergent series
E451513
The theory of divergent series is a branch of mathematical analysis that studies how to assign meaningful values to infinite series that do not converge in the usual sense, using specialized summation methods and analytic continuation.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hardy’s theory of divergent series | 1 |
| theory of divergent series canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4552203 — 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: theory of divergent series Context triple: [Orders of Infinity, hasPart, theory of divergent series]
-
A.
Dirichlet series
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
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B.
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.
-
C.
Lambert series
Lambert series are special infinite series in number theory and analysis, often involving arithmetic functions and powers of a variable, introduced by Johann Heinrich Lambert and used in the study of modular forms and q-series.
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D.
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.
-
E.
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.
- 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: theory of divergent series Target entity description: The theory of divergent series is a branch of mathematical analysis that studies how to assign meaningful values to infinite series that do not converge in the usual sense, using specialized summation methods and analytic continuation.
-
A.
Dirichlet series
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
-
B.
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.
-
C.
Lambert series
Lambert series are special infinite series in number theory and analysis, often involving arithmetic functions and powers of a variable, introduced by Johann Heinrich Lambert and used in the study of modular forms and q-series.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
branch of mathematical analysis
ⓘ
mathematical theory ⓘ |
| aimsTo |
extend the notion of sum of a series
ⓘ
provide consistent rules for assigning values to divergent series ⓘ relate divergent expansions to analytic functions ⓘ |
| appliesTo |
asymptotic power series in physics
ⓘ
formal series in differential equations ⓘ generating functions outside their domain of convergence ⓘ perturbation series in quantum field theory ⓘ power series with zero radius of convergence ⓘ |
| fieldOfStudy |
analytic continuation of series
ⓘ
asymptotic expansions ⓘ divergent series ⓘ generalized summation methods ⓘ resummation techniques ⓘ summation of divergent series ⓘ |
| hasApplication |
definition of path integrals via regularization
ⓘ
evaluation of asymptotic expansions in applied mathematics ⓘ regularization of divergent integrals ⓘ |
| hasHistoricalFigure |
Ernst Cesàro
ⓘ
G. H. Hardy ⓘ Leonhard Euler ⓘ Niels Henrik Abel ⓘ Srinivasa Ramanujan ⓘ Émile Borel ⓘ |
| hasKeyResult |
Hardy’s classification of summability methods
ⓘ
Tauberian theorems relating summability to convergence ⓘ conditions for equivalence of summation methods ⓘ |
| relatedTo |
asymptotic analysis
ⓘ
complex analysis ⓘ functional analysis ⓘ perturbation theory ⓘ quantum field theory ⓘ special functions ⓘ summability theory ⓘ |
| studies |
behavior of series outside their radius of convergence
ⓘ
infinite series that do not converge in the usual sense ⓘ methods to assign finite values to divergent series ⓘ relations between different summation methods ⓘ |
| usesConcept |
Abel summation
ⓘ
Borel summation ⓘ Borel transform ⓘ
linked to:
Borel summation
Cesàro summation ⓘ Euler summation ⓘ Ramanujan summation ⓘ Stokes phenomenon ⓘ analytic continuation ⓘ asymptotic series ⓘ resurgence theory ⓘ summability ⓘ zeta function regularization ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: theory of divergent series Description of subject: The theory of divergent series is a branch of mathematical analysis that studies how to assign meaningful values to infinite series that do not converge in the usual sense, using specialized summation methods and analytic continuation.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: theory of divergent series