Weierstrass M-test
E110608
convergence test
criterion for uniform convergence
theorem in complex analysis
theorem in real analysis
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Weierstrass M-test canonical | 1 |
| Weierstrass majorant test | 1 |
| Weierstrass uniform convergence test | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T940262 — 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: Weierstrass M-test Context triple: [Karl Weierstrass, notableFor, Weierstrass M-test]
-
A.
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.
-
B.
Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
-
C.
Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
-
D.
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
-
E.
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.
- 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: Weierstrass M-test Target entity description: 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.
-
A.
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.
-
B.
Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
-
C.
Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
-
D.
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
-
E.
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
convergence test
ⓘ
criterion for uniform convergence ⓘ theorem in complex analysis ⓘ theorem in real analysis ⓘ |
| appliesTo |
function series
ⓘ
series of functions ⓘ |
| assumes |
absolute bound on each term of the function series
ⓘ
convergence of the majorant numerical series ⓘ pointwise inequality between function terms and constants ⓘ |
| category | majorant test ⓘ |
| comparesTo | series of nonnegative constants ⓘ |
| concludes |
absolute convergence of the series of functions
ⓘ
uniform convergence of the series of functions ⓘ |
| domain |
metric spaces
ⓘ
normed vector spaces ⓘ |
| ensures |
limit function is continuous if each term is continuous
ⓘ
termwise differentiation is valid under additional hypotheses ⓘ termwise integration is valid on the domain ⓘ |
| field |
complex analysis
ⓘ
measure theory ⓘ
surface form:
real analysis
|
| generalizationOf | comparison test for numerical series ⓘ |
| hasAlternativeName |
Weierstrass M-test
ⓘ
surface form:
Weierstrass majorant test
Weierstrass M-test ⓘ
surface form:
Weierstrass uniform convergence test
|
| hasCondition |
existence of a sequence of nonnegative constants M_n
ⓘ
|f_n(x)| ≤ M_n for all x in the domain ⓘ ∑ M_n converges as a numerical series ⓘ |
| implies |
Cauchy criterion for the function series holds uniformly
ⓘ
sum of the function series is bounded by sum of M_n ⓘ |
| logicalStrength | sufficient but not necessary condition for uniform convergence ⓘ |
| namedAfter | Karl Weierstrass ⓘ |
| provides | sufficient condition for uniform convergence ⓘ |
| quantification | inequality holds for all points in the domain and all indices n ⓘ |
| relatedTo |
Fourier series
ⓘ
comparison test for series ⓘ power series convergence ⓘ uniform Cauchy criterion ⓘ |
| typeOfConvergence | uniform convergence ⓘ |
| typicalStatementForm | If |f_n(x)| ≤ M_n for all x and ∑ M_n converges, then ∑ f_n(x) converges uniformly ⓘ |
| usedFor |
establishing continuity of sums of function series
ⓘ
establishing uniform convergence on compact sets ⓘ interchanging limit and summation ⓘ justifying termwise differentiation ⓘ justifying termwise integration ⓘ |
| usedIn |
construction of holomorphic functions via series
ⓘ
proofs of uniform convergence of power series on compact subsets of the disk of convergence ⓘ theory of analytic functions ⓘ |
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: Weierstrass M-test Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Weierstrass uniform convergence test
this entity surface form:
Weierstrass majorant test