I'm interested in solving the equation $$ \color{red}{x}=1+\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\ddots \cfrac{a_n}{b_n+\color{red}{x}}}}, $$ where $a_i,b_i$ are positive real numbers. Is there a formula to simplify this continued fraction?
Any help will be appreciated.
$$b_{n-1}+\frac{a_n}{b_n+x}=\frac{b_{n-1}b_n+a_n+b_{n-1}x}{b_n+x}.$$
From this you can conclude that your equation has the form
$$x=\frac{px+q}{rx+s}$$ where the coefficients can be computed by recurrence. This equation can be rewritten as quadratic.