Sequences

A sequence is an ordered list of elements indexed by natural numbers. In mathematics, a sequence is often denoted by (a_n), where n = 1, 2, 3, ... and each a_n belongs to a given set, such as the real numbers. The nth term is the term of index n. Sequences can be finite or infinite, with infinite sequences the primary focus in analysis.

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. Sequences can be given explicitly, by a formula for a_n, or implicitly, via recurrence relations that define a_{n+1} in terms of earlier terms or n.

Convergence and limits: a sequence converges to a limit L if a_n approaches L as n grows without bound. If no such L exists, the sequence diverges. Divergence can take the form of oscillation, unbounded growth, or other behavior that prevents convergence. In the real numbers, a monotone sequence (nondecreasing or nonincreasing) that is bounded converges to its least upper bound or greatest lower bound, respectively.

Subsequences and compactness: a subsequence is obtained by selecting indices n_k with n_1 < n_2 < … . The Bolzano–Weierstrass theorem states that every bounded sequence of real numbers has a convergent subsequence.

Further topics include sequences of functions and notions of convergence (pointwise, uniform), as well as their role in series, continuity, and analysis more broadly.