We define the associated geometric series for a large class of positive linear operators and study the convergence of the series in the case of sequences of admissible operators. We obtain an inverse Voronovskaya theorem and we apply our results to the Bernstein operators, the Bernstein-Durrmeyer-type operators, and the symmetrical version of Meyer-König and Zeller operators.2000 Mathematics Subject Classification: 41A36; 41A27; 41A35. Key words: positive linear operators, iterates of operators, series of operators, antiderivative, inverse Voronovskaya operators, Bernstein operators, Bernstein-Durrmeyer type operators, symmetrical version of Meyer-König and Zeller operators.