Sequence: Difference between revisions

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imported>Catherine Woodgold
("indicating" rather than "showing", and punctuation)
imported>Catherine Woodgold
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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>.
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>.


A simple examples of sequences of the naturals, [[real numbers|reals]], or [[complex number]]s include (respectively)
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,....

Revision as of 08:17, 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 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}.

Basic definitions related to sequences

  • monotone sequences
  • subsequences
  • convergence of a sequence