Consider $\displaystyle f: \mathbb{C} \rightarrow \mathbb{C}$, $\displaystyle f(z) = z^2 - 2$.
Show that if $\displaystyle |z| > 2$, then the sequence z, f(z), f(f(z), ... diverges, i.e. is unbounded.
The sequence is alternatively defined by the recursive relation...
$\displaystyle a_{n+1} = a^{2}_{n}-2\ , \ a_{0}=z$ (1)
The procedure for solving (1) has been described a lot of times, the last in...
http://www.mathhelpforum.com/math-he...ce-176658.html
Following the procedure you find that...
a) if $\displaystyle |z|=|a_{0}|< 2 $ the sequence converges to -1
b) if $\displaystyle |z|=|a_{0}|= 2 $ the sequence converges to 2
c) if $\displaystyle |z|=|a_{0}|> 2 $ the sequence diverges...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$