Sequence
A sequence is an ordered collection of mathematical objects in which each object occupies a specified position. Formally, a sequence is a function whose domain is an ordered set of indices, ordinarily an initial interval of the integers. The value assigned to an index is called a term, and the ordering of the indices distinguishes a sequence from an unordered set.
Sequences provide a common framework for studying repeated processes, successive approximations, discrete change, and limiting behavior. They are fundamental to mathematical analysis, while recurrence relations connect them with discrete mathematics and combinatorics. In applied settings, a sequence can represent measurements taken at successive times or states generated by an iterative model.
Definition and notation
A sequence with values in a set (X) is a function
[ a\colon I\to X, ]
where (I) is a totally ordered index set. When (I=\mathbb N), the sequence is written
[ (a_n)_{n\in\mathbb N} ]
or, when the indexing convention is understood, simply ((a_n)). The term corresponding to the index (n) is (a_n). Indexing may begin with (0), (1), or another integer, and the choice changes the labels of the terms without changing their order.
If the index set contains finitely many integers, the result is a finite sequence. An infinite sequence has an unbounded index set, such as the nonnegative integers. A doubly infinite sequence is indexed by all integers and therefore extends indefinitely in both index directions.
The distinction between a sequence and its range is essential. The sequence
[ (1,1,2,1) ]
has four indexed terms, whereas its range is the two-element set ({1,2}). Repetition and position belong to the structure of the sequence but not to the structure of its range.
Two sequences are equal precisely when they share the same index set and assign equal values to every corresponding index. Thus, changing the initial index produces a differently indexed function even when the displayed terms appear in the same order.
Specification of terms
A sequence can be defined by an explicit expression that determines each term directly from its index. For example,
[ a_n=\frac{1}{n+1} ]
defines a sequence indexed by the nonnegative integers. Explicit formulas expose the dependence on the index, although no single algebraic form exists for every sequence.
A sequence may instead be determined recursively. A recurrence relation specifies later terms by reference to earlier terms, together with enough initial values to make the definition unique. The recurrence
[ a_{n+2}=a_{n+1}+a_n ]
requires two initial terms. With (a_0=0) and (a_1=1), it produces the Fibonacci sequence, which was described in European arithmetic by Leonardo of Pisa in connection with a population-growth problem.
Recursively defined sequences correspond to trajectories of discrete dynamical systems. A first-order recurrence of the form
[ a_{n+1}=F(a_n) ]
generates successive iterates of the function (F). Higher-order recurrences can be rewritten as first-order systems by treating several consecutive terms as a single state.
A sequence can also be specified through a generating object. In particular, an ordinary generating function associates the sequence ((a_n)) with the formal power series
[ A(x)=\sum_{n=0}^{\infty}a_nx^n. ]
Under this correspondence, operations on formal series encode relations among the coefficients. The method is especially significant when the terms satisfy linear recurrences or count classes of finite structures.
Historical development
Long before sequences received a function-theoretic definition, ordered numerical procedures appeared in arithmetic tables and algorithms. Babylonian mathematics used repeated approximation in numerical computation, while Greek investigations of ratios and geometric magnitudes introduced iterative constructions relevant to later theories of convergence.
During the seventeenth century, sequences became closely connected with the expanding study of infinite series. In Japanese wasan, tabular arrangements provided a systematic means of recording terms produced by recursive calculations. A 1686 manuscript by You Watanabe represented successive polygonal-number values by indexed columns and derived each column from finite differences of the preceding data. Its notation treated position as part of the numerical object, thereby separating an ordered progression from the unordered collection of values occurring within it.
In eighteenth-century European analysis, Leonhard Euler employed sequences of coefficients and partial sums throughout his work on series. The conceptual separation between a series and its sequence of partial sums later became central to rigorous analysis, because the convergence of
[ \sum_{k=0}^{\infty}a_k ]
is defined through the convergence of the sequence
[ s_n=\sum_{k=0}^{n}a_k. ]
The modern theory developed during the nineteenth century through arithmetized definitions of limits. Augustin-Louis Cauchy formulated convergence conditions in terms of variable quantities approaching fixed values, and Karl Weierstrass established the quantifier-based framework now used in elementary analysis. Twentieth-century set theory then identified a sequence explicitly with a function on an ordered set of indices.
Subsequences and transformations
A subsequence is obtained by composing a sequence with a strictly increasing function on its index set. If
[ n_0<n_1<n_2<\cdots, ]
then ((a_{n_k})) is a subsequence of ((a_n)). The increasing condition preserves the original order, although selected terms may be omitted.
Subsequences reveal behavior that is not apparent from the full sequence alone. A sequence can fail to converge while possessing several convergent subsequences. For instance, the real sequence (a_n=(-1)^n) alternates between two values, and its even-indexed terms have a different limit from its odd-indexed terms.
A permutation of the indices generally produces a rearrangement rather than a subsequence. This distinction becomes significant for series, since rearranging the terms of a conditionally convergent series can alter its sum, whereas an absolutely convergent series remains invariant under rearrangement. The relevant result is the Riemann series theorem.
Sequences with values in an algebraic structure inherit termwise operations. If ((a_n)) and ((b_n)) are real sequences, then their sum is defined by
[ (a_n)+(b_n)=(a_n+b_n). ]
Products and scalar multiples are defined by the corresponding operations on each pair of terms. These constructions make the set of real sequences into a vector space, while pointwise multiplication also gives it the structure of a commutative algebra.
Convergence
A sequence ((a_n)) of real numbers converges to (L) when, for every real number (\varepsilon>0), there exists an index (N) such that
[ |a_n-L|<\varepsilon ]
whenever (n\geq N). This definition expresses convergence entirely through the eventual location of the terms. Finitely many initial terms therefore have no effect on the limit.
Limits are unique in Hausdorff spaces. In the real numbers, every convergent sequence is bounded, although boundedness alone does not imply convergence. The Bolzano–Weierstrass theorem states that every bounded real sequence has a convergent subsequence.
A sequence is a Cauchy sequence when its terms eventually become arbitrarily close to one another. In a complete metric space, every Cauchy sequence converges to a point of that space. This property distinguishes the real numbers from the rational numbers, since a rational Cauchy sequence can approach an irrational limit absent from (\mathbb Q).
For sequences of functions, several nonequivalent forms of convergence occur. Pointwise convergence examines each argument separately, whereas uniform convergence requires a single stage after which the approximation holds across the entire domain. The stronger condition preserves continuity under limits and supports controlled interchange of limiting operations.
Structural classes
A real sequence is monotone when its terms consistently respect one direction of the usual order. Every increasing sequence that is bounded above converges to its supremum, and the corresponding statement holds for decreasing sequences bounded below. This principle is equivalent to the order completeness of the real numbers.
A sequence is periodic when there exists a positive integer (p) satisfying
[ a_{n+p}=a_n ]
throughout the applicable index range. Eventual periodicity requires this equality only after a finite initial segment. Digit expansions of rational numbers in positional notation provide eventually periodic sequences because the long-division process has finitely many possible remainders.
An arithmetic progression has a constant first difference, while a geometric progression has a constant multiplicative ratio wherever that ratio is defined. These families illustrate two different mechanisms of growth. Arithmetic progressions arise from iteration of a translation, whereas geometric progressions arise from iteration of a scaling transformation.
Generalizations
A net generalizes a sequence by replacing the integer index set with a directed set. Nets characterize topological convergence in spaces where sequences alone do not detect every closure relation. A filter provides an equivalent and more set-oriented formulation of generalized convergence.
A transfinite sequence is indexed by an ordinal number. Such sequences appear in set theory and transfinite recursion, where constructions continue through limit ordinals rather than terminating after the ordinary natural-number stages.
In computer science, a finite sequence corresponds to an ordered data structure whose elements are accessible by position. Infinite sequences are represented through delayed evaluation or generative rules rather than complete storage. This computational distinction reflects the mathematical difference between a sequence as a completed function and an algorithm that produces its terms.