Convergence of a reccurence defined sequence

Question

For which values of the parameter $a$ the following sequence is convergent?

$a_{n+1}=\frac{4a_{n}^2 - 4a_{n} + 2}{a_{n}+3}$ where $a_{1}=a$

I did it in the case when $a$ is bigger than $2$ but I can't reach an answer in the other cases.

Answer 1

Just a hint. The quickest way to find possible limits is to look at the fixed points of the function $f(x)=\frac{4x^2-4x+2}{x+3}$, i.e. points satisfying $f(x^*)=x^*$. There are $2$ such points $\left\{\frac{1}{3},2\right\}$:

• $\left|f'\left(\frac{1}{3}\right)\right|=\left|\frac{2 (2x^2+12x-7)}{(3 + x)^2}\right|_{x=\frac{1}{3}}=\frac{1}{2}<1$, so $x^*=\frac{1}{3}$ is attracting fixed point.
• $\left|f'\left(2\right)\right|=\left|\frac{2 (2x^2+12x-7)}{(3 + x)^2}\right|_{x=2}=2>1$, so $x^*=2$ is repelling fixed point.

From the theory, there is a vicinity $V$ of $x^*=\frac{1}{3}$ such that for any $a \in V$ we have $\lim\limits_{n\rightarrow\infty} a_n=\frac{1}{3}$.