Convergence

Convergence describes an indexed family approaching a limit: its terms become arbitrarily close to that limit as the index advances or meets a specified condition.

Limit of a sequence: A value that sequence terms approach as their indices increase. The standard numerical setting defines convergence by eventual closeness to this value.
Sequence: A function whose domain is an ordered set of indices, usually the natural numbers. Convergence tracks how the values of this indexed list behave as indices advance.
Divergent series: An infinite series whose sequence of partial sums does not approach a finite limit. Its partial sums fail the defining condition for convergence.
Series (mathematics): The sum of the terms of a sequence, defined through limits of finite partial sums. A series converges exactly when its partial sums approach a finite value.
Epsilon–delta definition: A formal definition of limits using arbitrary positive tolerances and corresponding bounds. It expresses convergence as eventual entry into every neighborhood of the proposed limit.
Limit: A value that a mathematical object approaches under a specified process. Convergence is the condition that an indexed family approaches such a value.
Bounded sequence: A sequence whose terms all lie within some fixed finite distance of a point. Boundedness alone does not require terms to approach any limit.
Power series: An infinite series whose terms are coefficients multiplied by powers of a variable. Its convergence determines where the represented function and its expansions are valid.
Metric space: A set equipped with a distance function satisfying the metric axioms. Distances let the same convergence definition apply beyond the real numbers.
Neighborhood (mathematics): A set containing an open set around a specified point. Convergence requires eventual membership in every neighborhood of the limit.

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