Euler's Constant, Generalized (I) - Definitions, Examples and Immediate Properties

It is known that the Euler-Mascheroni constant, which we will denote here by c0, plays a very important role in Mathematics. It is the limit of a decreasing sequence of real numbers, which we will denote here by cn or cn,0, a sequence that belongs to the interval (0,1), but which does not converge to 0. In this paper we propose to generalize this known constant c0 of Euler-Mascheroni, in positive and negative sense. In this sense, for each (0,1), starting from two generalizations of the sequence cn that converges to c0, we will obtain two sequences cn, and cn,-, which converge to c and c- respectively. We will call these limits, c - the positive generalized Euler-Mascheroni constant or the positive generalized Euler constant, respectively c- - the negatively generalized Euler-Mascheroni constant or the negatively generalized Euler constant. By calculating in two different ways the limits of some sequences, we will obtain the integral form of these two constants c and c-, and then we will calculate these two constants for different rational values of the number (0,1). Determining these generalized constants will allow one to easily compute many limits of sequences of real numbers that are otherwise much more difficult to compute. The present paper is meant to be one of Mathematics Didactics and comes to the support of teachers, students and pupils who want to train new skills and develop their competences for solving exercises and problems in Mathematics.