Infinite series: existence of power to ensure finiteness

Question

Let $(a_n)_{n\in\mathbb{N}}\subseteq[0,1]$ be a sequence converging to $0$. Does there always exist a real number $r\in[0,\infty)$ such that $\sum_{n\in\mathbb{N}}|a_n|^r<\infty$? Does such real number exist, for example, for the series $\sum_{n\in\mathbb{N}}|\frac{1}{log(n+1)}|^r$?