Efficiently determining Convergence in Polynomial Recurrence Sequences

­ We derive the necessary and sufficient condition, for a given Polynomial Recurrence Sequence to converge to a given target rational K. By converge, we mean that the Nth term of the sequence, is equal to K, as N tends to positive infinity. The basic idea of our approach is to construct a univariate polynomial equation in x, whose coefficients correspond to the terms of the Sequence. The approach then obtains the condition by analyzing five cases that cover all possible real values of x. The condition can be evaluated within time that is a polynomial function of the size of the description of the Polynomial Recurrence Sequence, hence convergence or non­convergence can be efficiently determined. There has been a lot of study [1][2][3] into the convergence properties of linear recurrence sequences and polynomial recurrence sequences. Some authors have focussed on whether the value of the N th term of the sequence (not necessary an integer sequence), can asymptotically converge to some real, as N tends to infinity. Other authors have focussed on whether the ratio of the N th term to the (N+1) th term can asymptotically converge to some real, as N tends to infinity. In this paper, we develop an approach to determine whether or not the N th term of a given sequence, can become equal to a given target rational K, as N tends to infinity. The starting points and the coefficients of the sequence, are rationals. The term " rational " denotes a real number (x/y) where both x and y are integers and y≠0. In the rest of this paper, when we say p i " converges to K " , we mean that the value of p N = K, as N tends to infinity. In this paper, by " infinity " we mean " positive infinity ". We also denote the absolute value function of x as abs(x), so abs(x) = x if x ≥ 0, and abs(x) = ­x if x < 0. Sections 2 and 3 apply our approach to homogeneous linear recurrence sequences and non­homogeneous linear recurrence sequences, respectively. Section 4 applies our approach to polynomial recurrence sequences 2. Homogeneous Linear Recurrence Sequences Consider a homogeneous linear recurrence sequence as follows: p i = c i , for all integers i in [0,L­1] , and p i = p i­1 a 1 + p i­2 a 2 + …