Tests for the convergence of continued fractions, based on the fundamental system of inequalities

S. S. Khloponin

Matematičeskie zametki, Tome 20 (1976) no. 5, pp. 665-674.

Résumé

We have proved that if the partial numerators of the continued fraction $f(c)=\frac11+\frac{c_2}1+\frac{c_3}1+\dots$ are all nonzero and for at least some number $n\ge1$ satisfy the inequalities $$ p_n|1+c_n+c_{n+1}|\ge p_{n-2}p_n|c_n|+|c_{n+1}|\quad(n\ge1,\quad p_{-1}=p_0=c_1=0,\quad p_n\ge0), $$ then $f(c)$ converges in the wide sense if and only if at least one of the series \begin{gather} \sum_{n=1}^\infty|c_3c_5\dots c_{2n-1}/(c_2c_4\dots c_{2n})|, \\ \sum_{n=1}^\infty|c_3c_4\dots c_{2n}/(c_3c_5\dots c_{2n+1})|. \end{gather}