Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an arbitrary index set.

"Sequential" redirects here. For the manual transmission, see Sequential manual transmission. For the sequentional logic function, see Sequention. For the American synthesizer company, see Sequential (company). For other uses, see Sequence (disambiguation).

For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6, ...).

The position of an element in a sequence is its rank or index; it is the natural number for which the element is the image. The first element has index 0 or 1, depending on the context or a specific convention. In mathematical analysis, a sequence is often denoted by letters in the form of $a_{n}$ , $b_{n}$ and $c_{n}$ , where the subscript n refers to the nth element of the sequence; for example, the nth element of the Fibonacci sequence $F$ is generally denoted as $F_{n}$ .

In computing and computer science, finite sequences are usually called strings, words or lists - with the specific technical term chosen depending on the type of object the sequence enumerates and the different ways to represent the sequence in computer memory. Infinite sequences are called streams.

The empty sequence ( ) is included in most notions of sequence. It may be excluded depending on the context.

$(1,9,25,\ldots )$

$(a_{1},a_{3},a_{5},\ldots ),\qquad a_{k}=k^{2}$

${(a_{2k-1})}_{k=1}^{\infty },\qquad a_{k}=k^{2}$

${(a_{k})}_{k=1}^{\infty },\qquad a_{k}=(2k-1)^{2}$

${\bigl (}(2k-1)^{2}{\bigr )}_{k=1}^{\infty }$

An is a sequence whose terms are integers.

integer sequence

A is a sequence whose terms are polynomials.

polynomial sequence

A positive integer sequence is sometimes called multiplicative, if a_nm = a_n a_m for all pairs n, m such that n and m are .^[8] In other instances, sequences are often called multiplicative, if a_n = na₁ for all n. Moreover, a multiplicative Fibonacci sequence^[9] satisfies the recursion relation a_n = a_n−1 a_n−2.

coprime

A is a sequence whose terms have one of two discrete values, e.g. base 2 values (0,1,1,0, ...), a series of coin tosses (Heads/Tails) H,T,H,H,T, ..., the answers to a set of True or False questions (T, F, T, T, ...), and so on.

binary sequence

$\lim _{n\to \infty }(a_{n}\pm b_{n})=\lim _{n\to \infty }a_{n}\pm \lim _{n\to \infty }b_{n}$

$\lim _{n\to \infty }ca_{n}=c\lim _{n\to \infty }a_{n}$ for all real numbers $c$

$\lim _{n\to \infty }(a_{n}b_{n})={\bigl (}\lim _{n\to \infty }a_{n}{\bigr )}{\bigl (}\lim _{n\to \infty }b_{n}{\bigr )}$

$\lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}={\bigl (}\lim \limits _{n\to \infty }a_{n}{\bigr )}{\big /}{\bigl (}\lim \limits _{n\to \infty }b_{n}{\bigr )}$ , provided that $\lim _{n\to \infty }b_{n}\neq 0$