  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 non-convergence can be efficiently
determined.
