explaining a recurrent series

Question

Let a>2 .

$\phi=a- \frac{1}{a- \frac{1}{a-\frac{1}{a-\frac{1}{....}}}}$

$\phi_0=a$ and $\phi_{n+1}=a-\frac{1}{\phi_n}$

How to explain the relation between $\phi_0, \phi_1, \phi_3$ series and $\phi$

Answer 1

Let $\phi=a-\dfrac{1}{\phi}$, then we have $$\phi_2=a-\dfrac{1}{a-\dfrac{1}{\phi}},$$

$$\phi_3=a-\dfrac{1}{a-\dfrac{1}{a-\dfrac{1}{\phi}}},$$ etc.

We can get $\phi$ from $\phi=a-\dfrac{1}{\phi}$, i.e. $$\phi^2-a\phi+1=0.$$

We have

$$\phi_{1,2}=\dfrac{a\pm \sqrt{a^2-4}}{2}.$$

Answer 2

This continue fraction is infinity. Thus $lim \phi_n\to \phi$. Note that $\phi=a-\frac{1}{\phi}$. You can find the roots of $ \phi^2- a \phi +1=0$ to find $\phi$.

Answer 3

If $\phi$ has any meaning at all, it is the limit of the sequence $\phi_n$. The challenge is to prove that the limit exists if $a > 2$.