# What is a sequence of real numbers?

An infinite sequence (or shorter a sequence) of real numbers is an application whose domain is an infinite subset of and whose range is .

Notations:

 Instead of we denote and we call this the sequence the term of index

Example 3.1.1

• Define a sequence as follows: the term with index (with ) is the digit in the decimal expansion of . We have: , , , , etc.
• The general term of can be given by an explicit formula, as in . Here we have: , , , , , .
• The general term of can be given by a recurrence formula, as in . In this case, the first term of the sequence has to be given.

Take . Then , , .

• The general term of can be given by a recurrence formula on two terms (or more). The most famous example is Fibonacci's sequence, denoted :

Then , , , , , , .

Definition 3.1.2   The sequence is periodic if there exists a natural number such that . The least such number is called the period of the sequence.

Example 3.1.3   Denote by the digit in the decimal development of the rational number ; the sequence is periodic and its period is equal to 6:

Noah Dana-Picard 2007-12-28