We can easily figure out that
where is the 'golden ratio' .
(Proof: is equal to plus its reciprocal, or . After multiplying both sides by and moving all of the terms to the left, we obtain
which can be evaluated to give the solution .)
One can see that a continued fraction of this form can be generalized for any continued fraction of the form
which has a solution . I will omit the derivation, which is easy to derive by modifying the proof above.
However, what about
What is the limit of this continued fraction as ? What is the limit as of any continued fraction of the form