Summability method

A procedure that assigns generalized sums to sequences or series that do not converge ordinarily. It extends ordinary summation while preserving familiar limits when they exist.

Cesàro summation: A summability method that assigns a sequence the limit of the averages of its successive partial sums. Averages can converge even when the original partial sums oscillate.
Convergence of a sequence: The property that sequence terms approach a finite limit as their indices increase. Ordinary convergence is the standard that a regular method must preserve.
Ordinary convergence: Convergence defined by the usual finite limit of a sequence or its partial sums. Summability extends this stricter notion rather than replacing it for convergent inputs.
Grandi's series: The divergent series 1 − 1 + 1 − 1 + …, whose partial sums alternate between one and zero. Cesàro and Abel procedures both assign it the value one-half.
Summability field: The set of sequences or series to which a given summability method assigns finite values. Comparing these domains shows which divergent inputs each method can handle.
Abel summation: A method that assigns a series the limit of its power series as its variable approaches one from below. Damping terms by powers creates a convergent expression before the boundary limit is taken.
Infinite series: A sum whose terms form an infinite sequence, defined through the limit of its partial sums when that limit exists. Summability methods extend the notion of a series sum beyond convergent partial sums.
Absolute convergence: Convergence of a series of absolute values, which guarantees convergence independent of term order. Absolute convergence needs no generalized assignment and is robust under rearrangement.
1 − 2 + 3 − 4 + …: The divergent series of alternating positive integers, with terms 1, −2, 3, −4, and so on. Abel summation assigns it one-quarter despite its unbounded partial sums.
Consistency of summability methods: The property that two methods assign the same value whenever both sum the same series. Consistency determines whether generalized sums are stable across procedures.

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