We consider series of the form pq+∑j=2∞1xj,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{p}{q} +\sum _{j=2}^\infty \frac{1}{x_j}, \end{aligned}$$\end{document}where x1=q\documentclass[12pt]{minimal}
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\begin{document}$$x_1=q$$\end{document} and the integer sequence (xn)\documentclass[12pt]{minimal}
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\begin{document}$$(x_n)$$\end{document} satisfies a certain non-autonomous recurrence of second order, which entails that xn|xn+1\documentclass[12pt]{minimal}
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\begin{document}$$x_n|x_{n+1}$$\end{document} for n≥1\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 1$$\end{document}. It is shown that the terms of the sequence, and multiples of the ratios of successive terms, appear interlaced in the continued fraction expansion of the sum of the series, which is a transcendental number.
