# Sequence

A **sequence**, in mathematics, is an enumerated list; the elements of this list are usually referred as to the *terms*. Sequences may be finite or infinite.

Formally, given any set *X*, an infinite sequence is a function (*f*, say) defined on the natural numbers , with values in *X*. Similarly, a finite sequence is a function *f* defined on with values in *X*. (We say that *n* is the *length* of the sequence).

In a natural way, the sequences are often represented as lists:

where, formally, , etc. Such a list is then denoted as , with the parentheses indicating the difference between the actual sequence and a single term .

Some simple examples of sequences of the natural, real, or complex numbers include (respectively)

- 10, 13, 10, 17,....
- 1.02, 1.04, 1.06,...

Often, sequences are defined by a general formula for . For example, the sequence of odd naturals can be given as

There is an important difference between the finite sequences and the [[set]s. For sequences, by definition, the order is significant. For example the following two sequences

- 1, 2, 3, 4, 5 and 5, 4, 1, 2, 3

are different, while the sets of their terms are identical:

- {1, 2, 3, 4, 5} = {5, 4, 1, 2, 3}.

Moreover, due to indexing by natural numbers, a sequence can list the same term more than once. For example, the sequences

- 1, 2, 3, 3, 4, 4 and 1, 2, 3, 4

are different, while for the sets we have

- {1, 2, 3, 3, 4, 4} = {1, 2, 3, 4}.

- monotone sequence
- subsequences
- convergence of a sequence