Abstract: This paper reveals the convergence nature of a logarithm item subtracted from harmonic series.A general method was extracted from this special case to be nominated as the Euler's convergence technique. Moreover,the Euler's convergence technique in many exhale series is used to construct new convergence series. Then,a set of positive real number is obtained and named as the generalized Euler'constant family. Finally,typical numerical experiments were given to show the computational results of the generalized Euler' constant family.