Abel summation

E451515

concept in mathematical analysis summability method summation method

Abel summation is a method in mathematical analysis for assigning values to certain divergent series by considering the limit of their power series as the variable approaches 1 from below.

Try in SPARQL Jump to: Statements Referenced by

Statements (43)

Predicate	Object
instanceOf	concept in mathematical analysis ⓘ summability method ⓘ summation method ⓘ
appliesTo	conditionally convergent series ⓘ divergent series ⓘ infinite series ⓘ
basedOn	limit of power series as variable approaches 1 from below ⓘ power series ⓘ
category	infinite series techniques ⓘ methods of summability ⓘ
characterizedBy	taking limit of sum a_n x^n as x approaches 1 from below ⓘ
contrastsWith	ordinary termwise convergence of series ⓘ
field	mathematical analysis ⓘ series summation ⓘ summability theory ⓘ
formalDefinition	A series ∑ a_n is Abel summable to s if lim_{x→1−} ∑ a_n x^n = s ⓘ
generalizes	ordinary sum of absolutely convergent series ⓘ
hasAlternativeName	Abelian summation NERFINISHED ⓘ
hasProperty	extends ordinary convergence of series ⓘ linear summation method ⓘ regular summation method ⓘ stable under scalar multiplication ⓘ stable under termwise addition ⓘ
implies	ordinary convergence when Abel sum exists and series is convergent ⓘ
introducedIn	19th century ⓘ
namedAfter	Niels Henrik Abel NERFINISHED ⓘ
relatedConcept	Abel transform NERFINISHED ⓘ Abel’s limit theorem NERFINISHED ⓘ
relatedTo	Abel’s theorem NERFINISHED ⓘ Borel summation ⓘ Cesàro summation ⓘ Tauberian theorems NERFINISHED ⓘ power series convergence ⓘ
requires	existence of limit of associated power series at x = 1− ⓘ
strongerThan	Cesàro summation of order 1 in many contexts ⓘ
toolFor	defining sums via analytic continuation of power series ⓘ regularization of divergent series ⓘ
usedFor	assigning values to divergent series ⓘ studying convergence of series ⓘ
usedIn	Fourier series theory NERFINISHED ⓘ analytic number theory ⓘ asymptotic analysis ⓘ study of generating functions ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Divergent Series → topic → Abel summation ⓘ