Cesàro summation
E451514
Cesàro summation is a method of assigning finite values to certain divergent series by averaging their partial sums.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Cesàro summation canonical | 4 |
| Cesàro mean of order α | 1 |
| Cesàro summation method | 1 |
| Cesàro-type averages | 1 |
| Fejér summation | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4552231 — 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: Cesàro summation Context triple: [Divergent Series, topic, Cesàro summation]
-
A.
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.
-
B.
Euler’s method of rearranging absolutely convergent series
Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
-
C.
Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
-
D.
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.
-
E.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
- 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: Cesàro summation Target entity description: Cesàro summation is a method of assigning finite values to certain divergent series by averaging their partial sums.
-
A.
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.
-
B.
Euler’s method of rearranging absolutely convergent series
Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
-
C.
Ramanujan’s sum
Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.
-
D.
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.
-
E.
Bernoulli numbers
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
method of summability ⓘ summation method ⓘ |
| appliesTo |
convergent series
ⓘ
divergent series ⓘ infinite series ⓘ series of complex numbers ⓘ series of real numbers ⓘ |
| basedOn | averaging partial sums of a series ⓘ |
| classification |
linear summation method
ⓘ
regular summation method ⓘ |
| definition | a series is Cesàro summable to L if the arithmetic means of its partial sums converge to L ⓘ |
| example | the Grandi series 1 - 1 + 1 - 1 + ... is Cesàro summable to 1/2 ⓘ |
| field |
mathematical analysis
ⓘ
series (mathematics) ⓘ summability theory ⓘ |
| generalization | (C,k) Cesàro summation of higher order ⓘ |
| generalizationOf | ordinary convergence of series via averaging of partial sums ⓘ |
| hasFormula | σ_n = (1/(n+1)) Σ_{k=0}^n s_k where s_k are partial sums ⓘ |
| hasOrder | first-order Cesàro method (C,1) ⓘ |
| hasVariant |
Cesàro mean of order α
ⓘ
linked to:
Cesàro summation
|
| implies | ordinary convergence when a series is convergent ⓘ |
| influenced | development of modern summability theory ⓘ |
| isWeakerThan | ordinary convergence of series ⓘ |
| namedAfter | Ernesto Cesàro ⓘ |
| notation | (C,1) summability ⓘ |
| property |
does not assign a value to every divergent series
ⓘ
every ordinarily convergent series is Cesàro summable to the same sum ⓘ preserves limits of convergent series ⓘ some divergent series are Cesàro summable ⓘ |
| purpose |
assign values to divergent series
ⓘ
generalize the notion of series convergence ⓘ |
| relatedConcept |
Fejér kernel
ⓘ
sequence transformation ⓘ summability matrix ⓘ |
| relatedTo |
Abel summation
ⓘ
Borel summation ⓘ Dirichlet series summation methods ⓘ
linked to:
Dirichlet series
Fejér summation ⓘ
linked to:
Cesàro summation
Hardy’s theory of divergent series ⓘ
linked to:
theory of divergent series
Tauberian theorems ⓘ |
| usedIn |
Fourier series
ⓘ
divergent series theory ⓘ harmonic analysis ⓘ |
| yearIntroduced | late 19th century ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Cesàro summation Description of subject: Cesàro summation is a method of assigning finite values to certain divergent series by averaging their partial sums.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Cesàro summation
linked to: Cesàro summation
linked to: Cesàro summation
linked to: Cesàro summation