with
alternates between 1 and -1.
Does f(n+1)/f(n) converge as n->infinity for f(n) defined by a linear recursion, for ALL linear recursions?
Edit: actually, I guess they do since they can all be expressed in an explicit formula and f(n+1)/f(n) expressed in terms of the explicit formula will yield a convergence. Right?
Considering the most general linear recurrence solution in explicit form:
with
and polynomials
I think the limit of the ratio converges, if for any terms in the explicit formula.
i.e. consider the roots to the characteristic equation with largest absolute value V. If both roots V and -V are present, and their coefficients are nonzero, then the ratios do not converge.