Convergence of a positive simple continued fraction

When $b_j=1$ and $a_j$ are positive integers, adjacent convergents satisfyEven and odd convergents are monotone from opposite sides, their separation is $1/(q_n{q}_{n+1})$, and $q_n\to \infty$. They therefore have a common limit $x$, with

If every $a_j=b_j=1$, then $p_n=F_{n+2}$ and $q_n=F_{n+1}$. Binet's formulashows that the convergents tend to $\varphi$ and gives sharp scaled errors through $F_{n+1}-\varphi F_n={\psi }^n$.