Sequences order terms so repetition and position both matter
Unlike a set, a mathematical sequence keeps order and allows the same object to appear more than once. Finite lists, infinite streams, Fibonacci numbers, primes, and decimal approximations of reals all count as sequences, and their convergence underpins series and much of analysis.
A sequence is an ordered collection of objects that may repeat. Membership alone does not determine it: (M, A, R, Y) differs from (A, R, M, Y), and (1, 1, 2, 3, 5, 8) is allowed because the first 1 appears twice. Sequences may be finite or infinite, as with the positive even integers 2, 4, 6, 8, and so on. Length counts elements of a finite sequence; each element's rank or index is the natural number whose image it is, often starting at 0 or 1. Computing calls finite sequences strings, words, or lists, and infinite ones streams. The empty sequence is usually included but sometimes excluded by context.
Analysts use sequences to probe functions and spaces via convergence, and series build on them for differential equations and analysis. Patterns such as primes or Fibonacci numbers are studied for their own sake. Listing works when the pattern is clear—the first four odd positives are (1, 3, 5, 7); primes begin (2, 3, 5, 7, 11, …); Fibonacci starts (0, 1, 1, 2, 3, 5, 8, …) with each term the sum of the previous two. Rational sequences can approach irrationals: .9, .99, .999, … tends to 1, and successive decimals of π increase toward π. Every real is a limit of rationals, for instance via its decimal expansion.
Elements need not be numbers: the monomials 1, x, x squared, x cubed, … form a sequence of functions. The On-Line Encyclopedia of Integer Sequences catalogs many integer examples. When listing is ambiguous, writers give a formula for the nth term with an index set—square numbers as (k squared) for k in the naturals, or (a_n) with a_n as a variable. Indices may run over naturals, a finite range such as k from 1 to 10, or all integers for a bi-infinite sequence extending both directions. When the index set is understood, subscripts are often omitted.
Recursive definitions contrast with closed formulas: a recurrence builds each term from earlier ones once enough initial terms are supplied. Multiple sequences can be tracked with different letters, and one can even form a sequence whose terms are themselves sequences. In short, sequences package ordered data so that limits, patterns, and series become speakable—the ordered list becomes the raw material of analysis rather than a mere enumeration.
Source: Sequence
