Cesàro summation
E451514
Cesàro summation is a method of assigning finite values to certain divergent series by averaging their partial sums.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Cesàro-type averages | 1 |
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 NERFINISHED ⓘ |
| implies | ordinary convergence when a series is convergent ⓘ |
| influenced | development of modern summability theory ⓘ |
| isWeakerThan | ordinary convergence of series ⓘ |
| namedAfter | Ernesto Cesàro NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
sequence transformation ⓘ summability matrix ⓘ |
| relatedTo |
Abel summation
ⓘ
Borel summation ⓘ Dirichlet series summation methods NERFINISHED ⓘ Fejér summation NERFINISHED ⓘ Hardy’s theory of divergent series NERFINISHED ⓘ Tauberian theorems NERFINISHED ⓘ |
| usedIn |
Fourier series
NERFINISHED
ⓘ
divergent series theory ⓘ harmonic analysis ⓘ |
| yearIntroduced | late 19th century ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Cesàro-type averages