<s>[INST] Is the sequence defined by $a_{n+1} = \frac{1}{a_n} + 1$ convergent? The sequence defined by $a_1=1$ and
 $$a_{n+1} = \frac{1}{a_n} + 1$$
is increasing. And I can get the limit which is equal to $$\frac{1+\sqrt{5}}{2}$$
But $a_2 = 2$ which is larger than the limit. So is the sequence convergent or divergent?
 [/INST] $\begin{array}\\
a_{n+1} 
&=1+ \frac{1}{a_n}\\
&=1+ \frac{1}{1+ \frac{1}{a_{n-1}}}\\
&=1+ \frac{1}{1+ \frac{1}{1+ \frac{1}{a_{n-2}}}}\\
&...\\
&=1+ \frac1{1+ \frac1{1+ \frac1{1+...}}}
\qquad n \text{ deep}\\
\end{array}
$
But that continued fraction
is
$\dfrac{F_{n+1}}{F_n}
\to \phi
$.
</s>