Dirichlet test
E466254
The Dirichlet test is a criterion in mathematical analysis that provides sufficient conditions for the convergence of certain infinite series, particularly those involving oscillatory terms.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Dirichlet test canonical | 1 |
| Dirichlet test for Fourier series | 1 |
| Dirichlet test for convergence | 1 |
| Dirichlet test for improper integrals | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4746244 — 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: Dirichlet test Context triple: [Peter Gustav Lejeune Dirichlet, notableWork, Dirichlet test]
-
A.
Cauchy condensation test
The Cauchy condensation test is a convergence criterion in mathematical analysis that determines whether an infinite series with positive, nonincreasing terms converges by comparing it to a related series formed by powers of two.
-
B.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
C.
Weierstrass M-test
The Weierstrass M-test is a criterion in real and complex analysis that provides a sufficient condition for the uniform convergence of a series of functions by comparing it to a convergent series of bounding constants.
-
D.
Dirichlet conditions
Dirichlet conditions are a set of sufficient criteria on a function—such as piecewise continuity and having a finite number of extrema and discontinuities on an interval—that guarantee the convergence of its Fourier series representation.
-
E.
Krak de l’Hospital
Krak de l’Hospital is an alternative name for Krak des Chevaliers, the famous medieval Crusader castle in Syria renowned for its massive fortifications and strategic importance.
- 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: Dirichlet test Target entity description: The Dirichlet test is a criterion in mathematical analysis that provides sufficient conditions for the convergence of certain infinite series, particularly those involving oscillatory terms.
-
A.
Cauchy condensation test
The Cauchy condensation test is a convergence criterion in mathematical analysis that determines whether an infinite series with positive, nonincreasing terms converges by comparing it to a related series formed by powers of two.
-
B.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
C.
Weierstrass M-test
The Weierstrass M-test is a criterion in real and complex analysis that provides a sufficient condition for the uniform convergence of a series of functions by comparing it to a convergent series of bounding constants.
-
D.
Dirichlet conditions
Dirichlet conditions are a set of sufficient criteria on a function—such as piecewise continuity and having a finite number of extrema and discontinuities on an interval—that guarantee the convergence of its Fourier series representation.
-
E.
Krak de l’Hospital
Krak de l’Hospital is an alternative name for Krak des Chevaliers, the famous medieval Crusader castle in Syria renowned for its massive fortifications and strategic importance.
- F. None of above. chosen
Statements (33)
| Predicate | Object |
|---|---|
| instanceOf |
concept in mathematical analysis
ⓘ
convergence test ⓘ mathematical criterion ⓘ |
| appliesTo |
oscillatory series
ⓘ
series of complex numbers ⓘ series of real numbers ⓘ |
| assumes |
b_n is monotone
ⓘ
limit of b_n equals 0 ⓘ sequence of partial sums of a_n is bounded ⓘ |
| category | tests for convergence of series ⓘ |
| concludes | series sum a_n b_n converges ⓘ |
| conditionOn |
partial sums of one factor are uniformly bounded
ⓘ
the other factor forms a monotone sequence tending to 0 ⓘ |
| contrastsWith | absolute convergence tests ⓘ |
| field |
infinite series
ⓘ
mathematical analysis ⓘ |
| generalizationOf | some special cases of alternating series convergence ⓘ |
| hasVariant |
Dirichlet test for improper integrals
ⓘ
linked to:
Dirichlet test
|
| historicalPeriod | 19th-century mathematics ⓘ |
| implies | conditional convergence when absolute convergence fails in some examples ⓘ |
| namedAfter | Peter Gustav Lejeune Dirichlet ⓘ |
| namedEntity | true ⓘ |
| provides | sufficient condition for convergence ⓘ |
| relatedTo |
Abel test
ⓘ
Fourier series ⓘ alternating series test ⓘ |
| strongerThan | alternating series test in some cases ⓘ |
| typicalForm | sum a_n b_n ⓘ |
| usedFor |
proving conditional convergence of series
ⓘ
proving convergence of Fourier series at a point ⓘ |
| usedIn |
analysis of Fourier coefficients
ⓘ
proofs of convergence for series with sine and cosine terms ⓘ theory of trigonometric series ⓘ |
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: Dirichlet test Description of subject: The Dirichlet test is a criterion in mathematical analysis that provides sufficient conditions for the convergence of certain infinite series, particularly those involving oscillatory terms.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Dirichlet test
linked to: Dirichlet test