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 $ \mathbb{N}$ and whose range is $ \mathbb{R}$ .

Notations:

Instead of we denote and we call this
\bgroup\color{blue}$ f$\egroup \bgroup\color{blue}$ (u_n)$\egroup the sequence
\bgroup\color{blue}$ f(n)$\egroup \bgroup\color{blue}$ u_n$\egroup the term of index \bgroup\color{blue}$ n$\egroup

Example 3.1.1       

Definition 3.1.2   The sequence $ (u_n)$ is periodic if there exists a natural number $ t \neq 0,1$ such that $ \forall n \in \mathbb{N}, \;
u_{n+t}=u_n$ . The least such number $ T$ is called the period of the sequence.

Example 3.1.3   Denote by $ u_n$ the $ n^{th}$ digit in the decimal development of the rational number $ \frac {11}{7}$ ; the sequence is periodic and its period is equal to 6:

$\displaystyle \frac {11}{7} = 1.571428571428571428...$    

Noah Dana-Picard 2007-12-28