Quote:
Originally Posted by RomanM
where r  real root of polynomial P(r), order >=6, and m, n  integers, and P(r+/eps)<1
P.S. I'm suspect that the simpler the question look like, the less likely it is to get an answer

It depends on what you're given first.
If r = x
_{1} is a Pisot number (an algebraic integer > 1 whose algebraic conjugates x
_{2},... x
_{n} all have absolute value less than 1) then for positive integer k, the sums
are all rational integers, and all the terms except the first tend to 0 as k increases without bound. Thus
becomes an increasingly good approximation as k increases.
The simplest case is with the polynomial P(x) = x^2  x  1. The sums are the Lucas numbers.
So the k
^{th} root of the k
^{th} Lucas number has limiting value equal to the root r > 1 of P(x) = 0.