# A Theorem on Divergence in the General Sense for Continued Fractions

## TL;DR

This paper establishes a theorem that clarifies the conditions under which continued fractions with differing odd and even part limits converge in the general sense, especially for certain classes of $q$ continued fractions.

## Contribution

It provides a comprehensive theorem on general convergence for a broad class of continued fractions with differing odd and even limits, and applies it to specific $q$ continued fractions.

## Key findings

- If $|q|>1$, the $q$ continued fraction either converges or does not converge in the general sense.
- Divergence of odd and even parts implies the growth of partial quotients.
- The theorem applies to two broad classes of $q$ 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 $\lim_{n \to \infty}|a_{n}| = \infty$.