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.

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All labels observed (3)

Label	Occurrences
Weierstrass M-test canonical	1
Weierstrass majorant test	1
Weierstrass uniform convergence test	1

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 ⓘ

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Instruction

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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.

Karl Weierstrass → notableFor → Weierstrass M-test ⓘ

Weierstrass M-test → hasAlternativeName → Weierstrass M-test ⓘ

this entity surface form: Weierstrass uniform convergence test

Weierstrass M-test → hasAlternativeName → Weierstrass M-test ⓘ

this entity surface form: Weierstrass majorant test