The Determinant Formula
It can be shown, by induction, that the determinant formula
holds for all positive integers n > 0. If neither Bn-1 nor Bn is zero, this relationship can also be used to express the difference between two successive convergents of the continued fraction.
It is necessary but not sufficient for the convergence of an infinite continued fraction that the difference between successive convergents approaches zero; this is the subject of the convergence problem. (Note: By definition, the continued fraction converges if the sequence of convergents has a limit.)
Read more about this topic: Fundamental Recurrence Formulas
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