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Math Help - Do ratio of consecutive terms converge for ALL linear recursions?

  1. #1
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    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?
    Last edited by oleholeh; December 27th 2010 at 08:46 AM.
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  2. #2
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    a_n = -a_{n-2} with a_0 = a_1 = 1

    a_{n+1}/a_n alternates between 1 and -1.
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  3. #3
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    D'oh. Then what conditions must hold?
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  4. #4
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    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.
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