  This paper deals with rates of convergence in the strong law of large
numbers, in the Baum-Katz form, for partial sums of Banach space valued random
variables. The results are then applied to solve similar problems for weighted
partial sums of conditional expectations. They are further used to treat
partial sums of powers of a reversible Markov chain operator. The method of
proof is based on martingale approximation. The conditions are expressed in
terms moments of the individual summands.
