Convergence problem

In the analytic theory of continued fractions, the convergence problem is the determination of conditions on the partial numerators ai and partial denominators bi that are sufficient to guarantee the convergence of the continued fraction In the analytic theory of continued fractions, the convergence problem is the determination of conditions on the partial numerators ai and partial denominators bi that are sufficient to guarantee the convergence of the continued fraction This convergence problem for continued fractions is inherently more difficult than the corresponding convergence problem for infinite series. When the elements of an infinite continued fraction consist entirely of positive real numbers, the determinant formula can easily be applied to demonstrate when the continued fraction converges. Since the denominators Bn cannot be zero in this simple case, the problem boils down to showing that the product of successive denominators BnBn+1 grows more quickly than the product of the partial numerators a1a2a3...an+1. The convergence problem is much more difficult when the elements of the continued fraction are complex numbers. An infinite periodic continued fraction is a continued fraction of the form where k ≥ 1, the sequence of partial numerators {a1, a2, a3, ..., ak} contains no values equal to zero, and the partial numerators {a1, a2, a3, ..., ak} and partial denominators {b1, b2, b3, ..., bk} repeat over and over again, ad infinitum. By applying the theory of linear fractional transformations to where Ak-1, Bk-1, Ak, and Bk are the numerators and denominators of the k-1st and kth convergents of the infinite periodic continued fraction x, it can be shown that x converges to one of the fixed points of s(w) if it converges at all. Specifically, let r1 and r2 be the roots of the quadratic equation These roots are the fixed points of s(w). If r1 and r2 are finite then the infinite periodic continued fraction x converges if and only if If the denominator Bk-1 is equal to zero then an infinite number of the denominators Bnk-1 also vanish, and the continued fraction does not converge to a finite value. And when the two roots r1 and r2 are equidistant from the k-1st convergent – or when r1 is closer to the k-1st convergent than r2 is, but one of the first k convergents equals r2 – the continued fraction x diverges by oscillation.