Cauchy’s integral test
E825422
Cauchy’s integral test is a convergence criterion in mathematical analysis that determines whether an infinite series converges by relating it to the behavior of a corresponding improper integral.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Cauchy’s integral test canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9843486 — 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: Cauchy’s integral test Context triple: [Augustin-Louis Cauchy, notableFor, Cauchy’s integral 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.
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B.
Dirichlet test
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.
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C.
Cauchy convergence criterion
The Cauchy convergence criterion is a fundamental concept in mathematical analysis that characterizes convergence of sequences (and series) by requiring that their terms become arbitrarily close to each other beyond some index.
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D.
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.
-
E.
Cauchy principal value
The Cauchy principal value is a method in mathematical analysis for assigning finite values to certain improper or divergent integrals and series by symmetrically balancing their singularities.
- 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: Cauchy’s integral test Target entity description: Cauchy’s integral test is a convergence criterion in mathematical analysis that determines whether an infinite series converges by relating it to the behavior of a corresponding improper integral.
-
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.
Dirichlet test
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.
-
C.
Cauchy convergence criterion
The Cauchy convergence criterion is a fundamental concept in mathematical analysis that characterizes convergence of sequences (and series) by requiring that their terms become arbitrarily close to each other beyond some index.
-
D.
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.
-
E.
Cauchy principal value
The Cauchy principal value is a method in mathematical analysis for assigning finite values to certain improper or divergent integrals and series by symmetrically balancing their singularities.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
convergence test
ⓘ
mathematical theorem ⓘ result in real analysis ⓘ |
| appearsIn |
textbooks on advanced calculus
ⓘ
textbooks on real analysis ⓘ |
| appliesTo |
infinite series
ⓘ
series of real numbers ⓘ |
| assumes |
f is continuous on [1,∞)
ⓘ
f is decreasing on [1,∞) ⓘ f is nonnegative on [1,∞) ⓘ monotone decreasing terms ⓘ positive terms ⓘ terms given by a function f(n) ⓘ |
| category |
theorem about improper integrals
ⓘ
theorem about infinite series ⓘ |
| compares |
∑ f(n)
ⓘ
∫₁^∞ f(x) dx ⓘ |
| concludes |
∑ 1/n^p converges for p>1 via integral of x^{-p}
ⓘ
∑ 1/n^p diverges for 0<p≤1 via integral of x^{-p} ⓘ |
| criterionType | convergence criterion ⓘ |
| field |
mathematical analysis
ⓘ
real analysis NERFINISHED ⓘ |
| formalizes | link between discrete sums and continuous integrals ⓘ |
| hasPrerequisite |
basic real analysis
ⓘ
knowledge of improper integrals ⓘ knowledge of infinite series ⓘ |
| historicalPeriod | 19th-century mathematics ⓘ |
| holdsFor | series with terms f(n)=1/n^p where p>0 ⓘ |
| implies |
if ∫₁^∞ f(x) dx is finite then ∑ f(n) is convergent
ⓘ
if ∫₁^∞ f(x) dx is infinite then ∑ f(n) is divergent ⓘ |
| logicalForm | if and only if statement between series and integral convergence under hypotheses ⓘ |
| namedAfter | Augustin-Louis Cauchy NERFINISHED ⓘ |
| relatedTo |
Cauchy condensation test
NERFINISHED
ⓘ
comparison test ⓘ integral comparison methods ⓘ p-series test ⓘ |
| relates |
improper integrals
ⓘ
series convergence ⓘ |
| requires | comparison between discrete sum and continuous integral ⓘ |
| states |
the series ∑ f(n) converges if the improper integral ∫₁^∞ f(x) dx converges
ⓘ
the series ∑ f(n) diverges if the improper integral ∫₁^∞ f(x) dx diverges ⓘ |
| usedFor |
testing convergence of p-series
ⓘ
testing convergence of series with algebraic terms ⓘ testing convergence of series with logarithmic factors ⓘ |
| usedIn |
calculus courses
ⓘ
problems involving asymptotic behavior of series ⓘ undergraduate analysis courses ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Cauchy’s integral test Description of subject: Cauchy’s integral test is a convergence criterion in mathematical analysis that determines whether an infinite series converges by relating it to the behavior of a corresponding improper integral.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Augustin-Louis Cauchy