This paper deals with rates of convergence in the strong law of largenumbers, in the Baum-Katz form, for partial sums of Banach space valued randomvariables. The results are then applied to solve similar problems for weightedpartial sums of conditional expectations. They are further used to treatpartial sums of powers of a reversible Markov chain operator. The method ofproof is based on martingale approximation. The conditions are expressed interms moments of the individual summands.