The rate of decay of $\sum_{j}(|a_{j} - T| + 1)^{-1} |a_{j}|$

Question

How do we find out the rate of decay of the series \[ \sum_{j}(|a_{j} - T| + 1)^{-q} |a_{j}|^{-1} \] as $T \to \infty$?

Suppose that $q > 0$ and \[ \sum_{j}|a_{j}|^{-1 - \epsilon} < \infty. \]

Any idea? Can we maybe analyze the sum case just by rewriting it as a suitable form?

Thanks for reading.