A Theorem on Divergence in the General Sense for Continued Fractions

Abstract

If the odd and even parts of a continued fraction converge to different values, the continued fraction may or may not converge in the general sense. We prove a theorem which settles the question of general convergence for a wide class of such continued fractions. We apply this theorem to two general classes of $q$ continued fraction to show, that if $G(q)$ is one of these continued fractions and $|q|>1$, then either $G(q)$ converges or does not converge in the general sense. We also show that if the odd and even parts of the continued fraction $K_{n=1}^{\infty }{a}_n/1$ converge to different values, then ${\lim }_{n\to \infty }|a_n|=\infty$.

11 pages

References in corpus (2)

• The Convergence and Divergence of $q$-Continued Fractions outside the Unit Circle
• The Convergence Behavior of $q$-Continued Fractions on the Unit Circle

Cited by in corpus (1)

• On the Divergence in the General Sense of $q$-Continued Fraction on the Unit Circle