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Epsilonn is a mathematical notational term used to denote a sequence of positive numbers, commonly written as ε_n, which tends to zero as n grows. In many texts, epsilonn is used to represent a family of shrinking error tolerances rather than a single fixed epsilon.

Usage and interpretation: In definitions of limits, convergence, and asymptotic analysis, one may encounter formulations such as |a_n - L| ≤ ε_n for all n beyond some index, with ε_n → 0. This emphasizes the rate at which the approximation improves as n increases. In numerical analysis and approximation theory, ε_n often bounds truncation or rounding errors that decrease with higher-order methods or finer discretization.

Notation and properties: There is no universal convention requiring ε_n to be monotone, though many authors assume monotone decrease to simplify arguments. The primary requirement is that ε_n is a sequence of nonnegative numbers converging to zero. As a result, ε_n can serve as a quantitative witness to convergence, giving a bound on how close a_n is to its limit for a given n.

Relation to other concepts: ε_n contrasts with a fixed ε in ε-δ definitions and in the common Big-O and little-o notations, where a nonnegative function of n bounds the error. In probability and statistics, ε_n may appear in definitions of convergence in probability with shrinking error bounds.

Etymology and see also: The symbol derives from the Greek letter epsilon, with the subscript n indicating a sequence. See epsilon, epsilon-delta, convergence, and asymptotic notation.