All-one continued fraction and Fibonacci ratios
= All-one continued fraction and Fibonacci ratios
If every $a_j=b_j=1$, then $p_n=F_{n+2}$ and $q_n=F_{n+1}$. Binet's formula
$$
F_n=\frac{\phi^n-\psi^n}{\sqrt5},
\qquad
\phi=\frac{1+\sqrt5}{2},
\quad
\psi=-\phi^{-1},
$$
shows that the convergents tend to $\phi$ and gives sharp scaled errors through $F_{n+1}-\phi F_n=\psi^n$.