Sequence: Difference between revisions
imported>Sébastien Moulin m (adding CZ Live category) |
imported>Michael Hardy |
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In a natural way, the sequences are often represented as lists: | In a natural way, the sequences are often represented as lists: | ||
:<math>a_1,\, a_2,\, a_3, | |||
:<math>a_1,\, a_2,\, a_3,\dots</math> | |||
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 making the difference between the actual sequence anda single term <math>a_n.</math> | Such a list is then denoted as <math>(a_n)</math>, with the parentheses making the difference between the actual sequence anda single term <math>a_n.</math> | ||
A simple examples of sequences of the naturals, [[real numbers|reals]] or [[complex | A simple examples of sequences of the naturals, [[real numbers|reals]], 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+''i'', 2-5''i'', 5-2''i''... | : 1 + ''i'', 2 - 5''i'', 5 - 2''i''... | ||
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 | ||
:<math> a_n= | |||
:<math> a_n = 2n + 1,\quad n=0,1,2,\dots</math> | |||
There is an important difference between the finite sequences and the [[set]s. | 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 | 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: | 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 | 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 | are different, while for the sets we have | ||
: {1, 2, 3, 3, 4, 4} = {1, 2, 3, 4}. | |||
==Basic definitions related to sequences== | ==Basic definitions related to sequences== |
Revision as of 14:34, 11 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 sequences 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