Solution to the monotonicity problem of an interesting class of sequences of real numbers

Abstract

Recently we have presented some sufficient conditions so that the real sequence (1+αn)n+p, n∈N, strictly increasingly converges to eα, when 0≤2p<α, and shown that the sequence is strictly decreasing if and only if 0<α≤2p. Here, we give a complete characterization of the monotonicity character of the sequence in the case 0≤2p<α, solving the most difficult case and the problem on the monotonicity character completely. To do this, among other things, we use several interesting new analytic inequalities.