The series approximations of ordinary differential equations usually possess a limited validity range. Outside the convergence interval, the solution does not comply with the real solution. A technique is proposed in this work to expand the validity interval of the series approximations. Solutions are constructed in the vicinity of several points and are joined to each other with satisfaction of the continuity conditions. By employing the piecewise series solutions, the validity range of the approximations can be made to cover the whole domain of interest. In constructing a single analytical expression from the piecewise series, the Gamma Interval function is used. Sample problems of first, second and third order are attacked with the proposed method and comparisons with the numerical and exact solutions are given. It is shown that the method can be effectively used in constructing approximate solutions of ordinary differential equations.