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Hi,
I have a sequence defined by a recurrence relation of the form
X(n) = f(n,1)*X(n-1)+ ... +f(n,k)*X(n-k)
Where the f's are k functions of n and each has a finite limit as n tends to
infinity.
I am interested in the asymptotic value of X(n)/X(n-1). What I have seen
done involves taking the recurrence and forming the characteristic
polynomial (where the f's are replaced by their respective limits) and
noting that the asymptotic value has to be one of the roots. What I can't
see, and a reference would be greatly appreciated, are what conditions on
the recurrence allow us to deduce that a finite asymptotic value exists at
all.
In the particular example I am thinking about the f's are quotients of small
order polynomials in n.
Regards,
Eric
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