We establish some properties of iterations of the remainder operator which assigns to any convergent series the sequence of its remainders. Moreover, we introduce the spaces of multiple absolute summable sequences. We also present some tests for multiple absolute convergence of series. These tests extend the well-known classical tests for absolute convergence of series. For example we generalize the Raabe, Gauss, and Bertrand tests. Next we present some applications of our results to the study of asymptotic properties of solutions of difference equations. We use the spaces of multiple absolute summable sequences as the measure of approximation. MSC: 39A10
Asymptotic properties of solutions of a difference equation of the form m x n = a n f (n, x σ (n)) + b n are studied. We present sufficient conditions under which, for any polynomial ϕ(n) of degree at most m-1 and for any real s ≤ 0, there exists a solution x of the above equation such that x n = ϕ(n) + o(n s). We give also sufficient conditions under which, for given real s ≤ m-1, all solutions x of the equation satisfy the condition x n = ϕ(n) + o(n s) for some polynomial ϕ(n) of degree at most m-1.
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