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 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,...
- 1 + i, 2 - 5i, 5 - 2i...
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 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}.
- monotone sequences
- subsequences
- convergence of a sequence