Solution
= Solution
For every positive integer $n$, the bound $|\sin(\omega n)|\le1$ gives
$$
|f(n)|=A n^p|\sin(\omega n)|\le A n^p.
$$
For $p<-1$, the positive <p-series> $\sum n^p$ converges. The <comparison test> therefore proves
$$
\boxed{\sum_{n=1}^\infty f(n)\text{ converges absolutely, and hence converges}.}
$$
No special arithmetic condition on $\omega$ is required for this sufficient condition.