Positive generalized continued fraction

For positive $a_j,b_j$, define $p_0=a_0$, $p_{-1}=1$, $q_0=1$, $q_{-1}=0$ andThenand $p_n/q_n$ is the corresponding finite generalized 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 $F_{n+1}-\varphi F_n={\psi }^n$.