 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""n""B""n"+1 grows more quickly than the product of the partial numerators "a"1"a"2"a"3..."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 {"a"1, "a"2, "a"3, ..., "a""k"} contains no values equal to zero, and the partial numerators {"a"1, "a"2, "a"3, ..., "a""k"} and partial denominators {"b"1, "b"2, "b"3, ..., "b""k"} repeat over and over again, "ad infinitum".