Some properties of the generalized Euler transformation (GET), such as the possibility of obtaining exact sum of a series, the convergence of a transformed series, and its new representations, are considered. Certain criteria of convergence of transformed series are presented and conditions, under which the Euler method enables one to obtain a finite expression for the sum of a series, are established. The properties of a transformed series are analyzed for the case that the known Pade, Pade-Borel, or Pade-Hermite approximants are used as the zero approximation in GET. Different ways of parameterization of the coefficients of the transformed series are discussed. The method proposed is tested while applied to the exactly solvable quantum-mechanics problem of the Kratzer oscillator taken as an example.