  Given a triangular array $\left\{X_{n,k}, \, 1 \leqslant k \leqslant n, n
\geqslant 1 \right\}$ of random variables satisfying $\mathbb{E} \lvert X_{n,k}
\rvert^{p} < \infty$ for some $p \geqslant 1$ and sequences $\{b_{n} \}$,
$\{c_{n} \}$ of positive real numbers, we shall prove that $\sum_{n=1}^\infty
c_n \mathbb{E} \left[ |\sum_{k=1}^n (X_{n,k} - \mathbb{E} \, X_{n,k})| / b_n -
\varepsilon \right]_+^p < \infty$, where $x_+ = \max(x,0)$. Our results are
announced in a general setting, allowing us to obtain the convergence of the
series in question under various types of dependence.
