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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
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This convergence problem for continued fractions is inherently more difficult than the corresponding convergence problem for infinite series.
== 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 ''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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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