Sequence: Difference between revisions
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Formally, given any set ''X'', an infinite sequence is a function (''f'', say) defined on | 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]] <math>\{1,2,3,...\}</math>, 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: | ||
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where, formally, <math>a_1=f(1)</math>, <math>a_2=f(2)</math> etc. | where, formally, <math>a_1=f(1)</math>, <math>a_2=f(2)</math> etc. | ||
Such a list is then denoted as <math>(a_n)</math>, with the parentheses | Such a list is then denoted as <math>(a_n)</math>, with the parentheses indicating the difference between the actual sequence and a single term <math>a_n</math>. | ||
Some simple examples of sequences of the natural, [[real numbers|real]], or [[complex number]]s include (respectively) | |||
: 10, 13, 10, 17,.... | : 10, 13, 10, 17,.... | ||
: 1.02, 1.04, 1.06,... | : 1.02, 1.04, 1.06,... | ||
: 1 + | : <math>1 + i, 2 + 3i, 3 + 5i,\dots</math> | ||
Often, sequences are defined by a general formula for <math>a_n</math>. For example, the sequence of odd naturals can be given as | Often, sequences are defined by a general formula for <math>a_n</math>. For example, the sequence of odd naturals can be given as | ||
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: 1, 2, 3, 4, 5 and 5, 4, 1, 2, 3 | : 1, 2, 3, 4, 5 and 5, 4, 1, 2, 3 | ||
are different, while the sets of | are different, while the sets of their terms are identical: | ||
: {1, 2, 3, 4, 5} = {5, 4, 1, 2, 3}. | : {1, 2, 3, 4, 5} = {5, 4, 1, 2, 3}. | ||
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==Basic definitions related to sequences== | ==Basic definitions related to sequences== | ||
*monotone | *[[monotone sequence]] | ||
*subsequences | *subsequences | ||
*convergence of a sequence | *convergence of a sequence | ||
Latest revision as of 15:56, 12 November 2008
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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