Sequence

In mathematics, a sequence is a list (family) of finitely or infinitely many consecutively numbered objects (for example, numbers). The same object can occur several times in a sequence. The object with the number , one says here also: with the index , is called -th member or -th component of the sequence. Finite as well as infinite sequences are found in all areas of mathematics. Infinite sequences, whose members are numbers, are mainly dealt with in calculus.

If is the number of members of a finite sequence, it is called a sequence of length , an -membered sequence, or an -tuple. The sequence without members, whose index range is empty, is called empty sequence, 0-membered sequence or 0-tuple.

Examples

$(1,0,0,2,1)$ 5-tuples of integers

$(\sin ,\ \cos ,\ \tan ,\ \cot )$ 4-tuples of trigonometric functions

$(2,3,5,7,11,13,\dotsc )$ Sequence of prime numbers

$(\{\},\{1\},\{1,2\},\{1,2,3\},\dotsc )$ Infinite sequence of sets.

$(x_{0},x_{1},x_{2},x_{3},\dotsc )$ General infinite sequence whose terms are continuously indexed. Here, zero is chosen as the indexing start.

Curves of the first 5 members of the sequence of functions $f_{n}(x)={\tfrac {x^{2}}{n}}$

Notation

In general, for a finite sequence writes $\left(a_{i}\right)_{i=1,\dots ,n}$ , so $(a_1,a_2,\dotsc,a_n)$ , and for infinite sequences $\left(a_{i}\right)_{{i\in {\mathbb N}}}$ , so $(a_1,a_2,\dotsc)$ . The $a_{i}$ represents any sequence member; the round brackets combine them into a sequence, then the running range of the index is shown (this may be omitted if it is implicitly clear). Pointed brackets are sometimes used instead of round brackets (i.e. ⟨ $\left\langle a_{i}\right\rangle _{i}$ ); semicolons may be used instead of commas if there is a risk of confusion with the decimal separator.

The difference from the set of sequence members $\lbrace a_i \mid i \in \N\rbrace$ or $\left\lbrace a_{i}\right\rbrace _{{i\in {\mathbb N}}}$ is that the order of $a_{n}$ matters and that several sequence members can have the same value.

Example: the sequence (0, 1, 0, 2, 0, 4, 0, 8, ...) has the image set (or underlying set) {0, 1, 2, 4, 8, ...}. The sequence (1, 0, 2, 0, 0, 4, 0, 0, 0, 8, ...) has the same image set. In both sequences the value 0 occurs several times.

Questions and Answers

Q: What is a sequence?

A: A sequence is a set of related events, movements or items that follow each other in a particular order.

Q: How is it used?

A: It is used in mathematics and other disciplines. In ordinary use, it means a series of events, one following another.

Q: What are two kinds of sequences?

A: The two kinds of sequences are finite sequences, which have an end, and infinite sequences, which never end.

Q: Can you give an example of an infinite sequence?

A: An example of an infinite sequence is the sequence of all even numbers bigger than 0. This sequence never ends; it starts with 2, 4, 6 and so on.

Q: How can we write down an infinite sequence?

A: We can write down an infinite sequence by writing a rule for finding the thing in any place one wants. The rule should tell us how to get the thing in the n-th place where n can be any natural number.

Q: What does (a_n) stand for when writing down a sequence?

A:(a_n) stands for the n-th term of the sequence.