On the convergence of series of moments for row sums of random variables

Given a triangular array $\left{X_{n,k}, , 1 \leqslant k \leqslant n, n \geqslant 1 \right}$ofrandomvariablessatisfying$$\mathbb{E}$ \lvert X_{n,k} \rvert^{p} < \infty$forsome$p \geqslant 1$andsequences${b_{n} }$,${c_{n} }$ofpositiverealnumbers,weshallprovethat$\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.