On analogue of Eiler-Maskeroni constant and regularities of its change

The paper is devoted to the description of approximation regularities of the partial sum of the generalized harmonic number series. Series of this type are often used in number theory, in the description of physical laws and computational mathematics. An option of the analytical estimation of the partial sum of the generalized harmonic series in the Eulerian form is proposed. That is, the partial sum is equal to the value of the general term integral of the series for the last counted term, plus a constant, plus an infinitesimal function.