Järjend

Järjend is the Estonian term for the mathematical concept known in English as a sequence. It is an ordered collection of elements indexed by natural numbers, where the position in the order determines each term. Formally, a sequence (a_n) assigns to every n in the natural numbers an element a_n from a set X. The nth term is denoted a_n, and sequences can be finite (ending after a finite number of terms) or infinite.

In notation, a sequence is often written as a1, a2, a3, ..., with the indexing starting at 1 or 0 depending on convention. Commonly studied sequences include arithmetic progressions, where a_n = a1 + (n−1)d, and geometric progressions, where a_n = a1 r^(n−1). Another famous example is the Fibonacci sequence, defined by F1 = F2 = 1 and F_n = F_{n−1} + F_{n−2} for n ≥ 3.

A central topic in analysis is the limit of a sequence. A sequence (a_n) converges to L if, for every ε > 0, there exists N such that |a_n − L| < ε for all n ≥ N. If no such L exists, the sequence diverges. Sequences may also be monotone (nondecreasing or nonincreasing) and bounded, properties that influence convergence and behavior.

Sequences underpin much of calculus and analysis, serving to define limits, continuity, and the construction of functions. They are also used in computer science, data analysis, and numerical methods, where they may be viewed as lists or arrays of values. A related concept is a series, the sum of the terms of a sequence.