Almost everywhere convergence of series in $L^1$

We answer positively a question of J. Rosenblatt (1988), proving the existence of a sequence $(c_i)$ with $\sum_{i=1}^{\infty}\vert c_i\vert=\infty$, such that for every dynamical system $(X,\Sigma, m, T)$ and $f\in L^1(X)$, $\sum_{i=1}^{\infty}c_i f(T^ix)$ converges almost everywhere. A similar result is obtained in the real variable context.