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

• Dec 27th 2010, 07:23 AM
oleholeh
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?
• Dec 27th 2010, 08:12 AM
snowtea
\$\displaystyle a_n = -a_{n-2}\$ with \$\displaystyle a_0 = a_1 = 1\$

\$\displaystyle a_{n+1}/a_n\$ alternates between 1 and -1.
• Dec 27th 2010, 08:44 AM
oleholeh
D'oh. Then what conditions must hold?
• Dec 27th 2010, 09:31 AM
snowtea
Considering the most general linear recurrence solution in explicit form:

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

I think the limit of the ratio converges, if \$\displaystyle 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.