Sequence

A sequence 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 a subset of natural numbers with values in X. Similarly, a finite sequence is a function f defined on $\{1,2,3,...,n\}$ 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:

$a_{1},\,a_{2},\,a_{3},\dots$

where, formally, $a_{1}=f(1)$, $a_{2}=f(2)$ etc. Such a list is then denoted as $(a_{n})$, with the parentheses indicating the difference between the actual sequence and a single term $a_{n}$.

A simple examples of sequences of the naturals, reals, or complex numbers include (respectively)

10, 13, 10, 17,....

1.02, 1.04, 1.06,...

1 + i, 2 - 5i, 5 - 2i...

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

$a_{n}=2n+1,\quad n=0,1,2,\dots$

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 its 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}.

Basic definitions related to sequences

• monotone sequences
• subsequences
• convergence of a sequence