  Let $\{X_n\}_{n\geq 1}$ be either a sequence of arbitrary random variables,
or a martingale difference sequence, or a centered sequence with a suitable
level of negative dependence. We prove Baum-Katz type theorems by only assuming
that the variables $X_n$ satisfy a uniform moment bound condition. We also
prove that this condition is best possible even for sequences of centered,
independent random variables. This leads to Marcinkiewicz-Zygmund type strong
laws of large numbers with estimate for the rate of convergence.
