# Thread: Do ratio of consecutive terms converge for ALL linear recursions?

1. ## Do ratio of consecutive terms converge for ALL linear recursions?

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?

2. $a_n = -a_{n-2}$ with $a_0 = a_1 = 1$

$a_{n+1}/a_n$ alternates between 1 and -1.

3. D'oh. Then what conditions must hold?

4. Considering the most general linear recurrence solution in explicit form:

$C_0P_0(n)r_0^n + C_1P_1(n)r_1^n + ...$
with $|r_0| \geq |r_1| \geq ...$
and $P_0, P_1, ...$ polynomials

I think the limit of the ratio converges, if $r_i \neq -r_0$ 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.