We establish convergence results involving the nth Cesaro mean σβn(θ) of order β of f(eiθ) = [sum ]aneinθ where the coefficients (an) satisfy a Dirichlet-type condition [sum ]nα[mid ]an[mid ]2 < ∞ for a fixed α in (0, 1]. When α = 1, for example, we prove that [sum ][mid ]σβn(θ)−f(eiθ) [mid ]2 < ∞ for every β > −½ at almost all points θ in [−π,π]. We also derive weighted convergence results of this type outside exceptional sets of capacity zero, and show by examples that these results are, in a certain sense, sharp with respect to the size of the exceptional sets.