A COUNTEREXAMPLE TO A CONTINUED FRACTION CONJECTURE

Abstract

It is known that if $a\in\mathbb{C}\setminus(-\infty,-\tfrac14]$ and $a_n\to a$ as $n\to\infty$, then the infinite continued fraction with coefficients $a_1,a_2,\dots$ converges. A conjecture has been recorded by Jacobsen et al., taken from the unorganized portions of Ramanujan’s notebooks, that if $a\in(-\infty,-\tfrac14)$ and $a_n\to a$ as $n\to\infty$, then the continued fraction diverges. Counterexamples to this conjecture for each value of $a$ in $(-\infty,-\tfrac14)$ are provided. Such counterexamples have already been constructed by Glutsyuk, but the examples given here are significantly shorter and simpler.