SEEMOUS 2021 Problem 4

For $p \in \mathbb{R}$ , let $(a_n)_{n \ge 1}$ be the sequence defined by $$a_n=\frac{1}{n^p} \int_0^n |\sin( \pi x)|^x \mathrm dx.$$ Determine all possible values of $p$ for which the series $\sum_{n=1}^\infty a_n$ converges.