We study power series of members of a class of positive linear operators reproducing linear function constituting a link between genuine Bernstein-Durrmeyer and classical Bernstein operators. Using the eigenstructure of the operators we give a non-quantitative convergence result towards the inverse Voronovskaya operators. We include a quantitative statement via a smoothing approach. Keywords: Power series geometric series positive linear operator Bernsteintype operator genuine Bernstein-Durrmeyer operator degree of approximation eigenstructure moduli of continuity. MSC 2010: 41A10 41A17 41A25 41A36