sequence

In mathematics, a sequence is an ordered list of objects indexed by the natural numbers. The nth term is written a_n, and a sequence is often denoted (a_n)_{n=1}^\infty. Sequences may be finite or infinite and are used to study patterns, limits, and recurrences.

Common examples include arithmetic sequences, where a_n = a_1 + (n−1)d; geometric sequences, where a_n = a_1 r^{n−1}; and the Fibonacci sequence, defined by a_1 = 1, a_2 = 1, and a_n = a_{n−1} + a_{n−2} for n > 2.

A central concept is convergence. An infinite sequence of real numbers (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 no such L exists, the sequence diverges. Other properties include monotonicity (nondecreasing or nonincreasing) and boundedness.

Every sequence can be viewed as a function from the natural numbers to a target set X. A subsequence is formed by selecting indices n_1 < n_2 < … and considering (a_{n_k}). The concept of convergence generalizes to metric and topological spaces.

A related concept is the series, which is the sum of the terms of a sequence. The partial sums form another sequence and are studied for convergence of the series.

Beyond mathematics, sequences appear in other fields, such as computer science (arrays and lists), biology (DNA sequences), and statistics (time series).