Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients

For complex linear homogeneous recursive sequences with constant coefficients we find a necessary and sufficient condition for the existence of the limit of the ratio of consecutive terms, characterizing this way the linear recursive sequences which behave as geometric sequences at infinity. The result can be applied even if the characteristic polynomial has not necessarily roots with modulus pairwise distinct, as in the celebrated Poincare's theorem. In case of existence, we characterize the limit as a particular root of the characteristic polynomial, which depends on the initial conditions and that is not necessarily the unique root with maximum modulus and multiplicity. The result extends to a quite general context the way to find the Golden mean as limit of ratio of consecutive terms of the classical Fibonacci sequence.