Sequence: Difference between revisions
imported>Michael Hardy |
imported>Catherine Woodgold m (singular; punctuation) |
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A '''sequence''' is an enumerated list; the elements of this list are usually referred as to the ''terms''. Sequences may be finite or infinite. | 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 | 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 <math>\{1,2,3,...,n\}</math> 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: | In a natural way, the sequences are often represented as lists: |
Revision as of 08:13, 28 April 2007
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 making the difference between the actual sequence anda single term
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 . 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