$ f$ is an affine function.

There exists a real number $ \alpha$ such that the sequence $ (b_n)$ determined by $ \forall n \in \mathbb{N}, b_n=a_n+ \alpha$ is geometric.

Example 3.9.1   Let $ (a_n)$ be the sequence of real numbers defined by its first term $ a_0=2$ and the recurrence relation $ a_{n+1}= \frac 23 a_n +4$ . We look for a number $ \alpha$ such that the announced sequence $ (b_n)$ will be geometric.

We have:

$\displaystyle b_{n+1}$ $\displaystyle = a_{n+1} + \alpha$    
$\displaystyle \quad$ $\displaystyle = \frac 23 a_n + 4 + \alpha$    
$\displaystyle \quad$ $\displaystyle = \frac 23 (b_n - \alpha) +4 +\alpha$    
$\displaystyle \quad$ $\displaystyle = \frac 23 b_n + \frac 13 \alpha +4$    

The sequence $ (b_n)$ is geometric if, and only if, $ \frac 13 \alpha +4 =0$ , i.e. $ \alpha = -12$ .

Such a computation enables us to decide whether the sequence $ (a_n)$ is convergent or not. Here the sequence $ (b_n)$ is a geometric sequence whose ratio is equal to $ \frac 23$ , thus it is convergent and its limit is 0. By Thm 7.1, we conclude that $ (a_n)$ is convergent and its limit is equal to 12.

Noah Dana-Picard 2007-12-28