Existence of the Limit of Ratios of Consecutive Terms for a Class of Linear Recurrences

Abstract

Let (Fn)n=1∞ be the classical Fibonacci sequence. It is well known that the limFn+1/Fn exists and equals the Golden Mean. If, more generally, (Fn)n=1∞ is an order-k linear recurrence with real constant coefficients, i.e., Fn=∑j=1kλk+1−jFn−j with n>k, λj∈R, j=1,…,k, then the existence of the limit of ratios of consecutive terms may fail. In this paper, we show that the limit exists if the first k elements F1,F2,…,Fk of (Fn)n=1∞ are positive, λ1,…,λk−1 are all nonnegative, at least one being positive, and max(λ1,…,λk)=λk≥k. The limit is characterized as fixed point, bounded below by λk and bounded above by λ1+λ2+⋯+λk.