Convergence problem

In the analytic theory of continued fractions, the convergence problem is the determination of conditions on the partial numerators a_i and partial denominators b_i 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 B_n cannot be zero in this simple case, the problem boils down to showing that the product of successive denominators B_nB_n+1 grows more quickly than the product of the partial numerators a₁a₂a₃...a_n+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, ..., a_k} contains no values equal to zero, and the partial numerators {a₁, a₂, a₃, ..., a_k} and partial denominators {b₁, b₂, b₃, ..., b_k} repeat over and over again, ad infinitum.

By applying the theory of linear fractional transformations to

where A_k-1, B_k-1, A_k, and B_k 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

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