Elementary Results
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.
Read more about this topic: Convergence Problem
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