Continued fraction tails and irrationality (Q915759)

The tails of a continued fraction \(K^{\infty}_{k=1}\frac{a_ k}{b_ k}\) are \(T_ k=K^{\infty}_{j=k}\frac{a_ j}{b_ j}\). Working with tails, the author proves the following theorem. Let sequences of positive integers \(a_ k\), \(b_ k\), \(P_{2k-1}\) for \(k\geq 1\) be given such that \(a_{2k-1}=b_{2k-1}P_{2k-1}+1\), \(b_{2k}\geq a_{2k}P_{2k-1}- P_{2k+1}\), and \(\sum^{\infty}_{k=1}P^{-1/2}_{2k-1}\) diverges. Then the continued fraction \(K^{\infty}_{k=1}\frac{a_ k}{b_ k}\) converges. Its value is irrational if and only if the inequality for \(b_{2k}\) is strict for infinitely many k.