Sequence

From Citizendium
Revision as of 04:46, 1 April 2007 by imported>Sébastien Moulin (adding CZ Live category)
Jump to navigation Jump to search

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}.


Basic definitions related to sequences

  • monotone sequences
  • subsequences
  • convergence of a sequence