järjendeid

Järjendeid, Estonian mathematics terminology for sequences, denote functions from the natural numbers to a set, typically the real numbers. The n-th term is written a_n, and the index n is usually taken from 1 or 0. Sequences describe ordered lists of elements and are distinct from series, which are sums of sequence terms.

Common examples include arithmetic sequences, where a_n = a + (n−1)d and consecutive terms differ by a constant d; and geometric sequences, where a_n = a r^{n−1} and consecutive terms differ by a constant ratio r. Sequences may be monotone (increasing or decreasing) or nonmonotone, and may be bounded or unbounded.

A central concept is convergence. A sequence (a_n) converges to a limit L if, for every ε > 0, there exists N such that |a_n − L| < ε for all n ≥ N. If a sequence has a finite L, it is convergent; otherwise it diverges. Related ideas include the Cauchy criterion: a sequence converges in the real numbers if and only if, for every ε > 0, there exists N such that |a_m − a_n| < ε for all m,n ≥ N. In R, every monotone and bounded sequence converges. Geometric sequences converge when |r| < 1, with the limit determined by the starting value and the ratio.

Subsequences are formed by selecting terms with increasing indices and can converge even when the original sequence does not. The distinction between a sequence and a series is key: a series ∑ a_n sums the terms of a sequence.