Cauchy condensation test
E239293
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Cauchy condensation test canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T2171654 — 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 condensation test Context triple: [Augustin-Louis Cauchy, knownFor, Cauchy condensation test]
-
A.
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.
-
B.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
-
C.
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.
-
D.
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.
-
E.
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.
- 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 condensation test Target entity description: 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.
-
A.
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.
-
B.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
convergence test
ⓘ
mathematical criterion ⓘ theorem in real analysis ⓘ |
| appliesTo |
infinite series
ⓘ
series with nonincreasing terms ⓘ series with positive terms ⓘ |
| appliesWhen | terms decrease sufficiently regularly ⓘ |
| assumes | index set is the positive integers ⓘ |
| attributedTo | Augustin-Louis Cauchy ⓘ |
| category | series convergence test ⓘ |
| compares |
condensed series
ⓘ
original series ⓘ |
| conditionOnTerms |
terms must be nonincreasing
ⓘ
terms must be nonnegative ⓘ |
| convergenceCriterion | sum a_n converges iff sum 2^n a_{2^n} converges ⓘ |
| definesCondensedSeries | sum of 2^n a_{2^n} ⓘ |
| doesNotApplyTo | series with sign-changing terms without modification ⓘ |
| field |
mathematical analysis
ⓘ
real analysis ⓘ |
| helpsShow |
divergence of harmonic series
ⓘ
growth rate of partial sums for some series ⓘ |
| implication |
sum 1/n^p converges for p>1
ⓘ
sum 1/n^p diverges for p<=1 ⓘ |
| language | stated in terms of sequences and series ⓘ |
| logicalForm | biconditional between convergence of two series ⓘ |
| namedAfter | Augustin-Louis Cauchy ⓘ |
| originalSeriesForm | sum of a_n from n=1 to infinity ⓘ |
| proofTechnique |
comparison of grouped sums with condensed series
ⓘ
grouping terms in blocks of dyadic length ⓘ |
| relatedTo |
comparison test
ⓘ
integral test ⓘ p-series test ⓘ |
| requires | monotone decreasing sequence of terms ⓘ |
| resultType | necessary and sufficient condition for convergence ⓘ |
| termType | real nonnegative terms ⓘ |
| typicalExample | series sum 1/n^p ⓘ |
| usedBy |
mathematicians studying series
ⓘ
students of analysis ⓘ |
| usedFor |
analyzing series with slowly decreasing terms
ⓘ
testing convergence of series similar to p-series ⓘ |
| usedIn |
calculus courses
ⓘ
undergraduate real analysis courses ⓘ |
| usesTransformation |
dyadic subsequence of terms
ⓘ
powers of two in the index ⓘ |
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: Cauchy condensation test Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Augustin-Louis Cauchy