TITLE: Does the ratio of consecutive terms converge for all linear recursions?
QUESTION [6 upvotes]: Does $f(n+1)/f(n)$ converge as $n\rightarrow\infty$ for $f(n)$ defined by a linear recursion, for all linear recursions?

REPLY [3 votes]: As Ross's counterexample points out the answer, of course, is no in general.  However there are broad special cases where the answer is yes, which all more or less go back to the Perron-Frobenius theorem.  In particular if the recurrence is combinatorial in the sense that it counts the number of words of length $n$ in a regular language, then subject to some reasonable assumptions about the language the limit you are looking at will exist.