Is the Series Sin(k)/k Absolutely Convergent?

We need to investigate whether $\sum_{k=1}^\infty \frac{|\sin k|}{k}$ convergent.

Note $\forall k\in\mathbb{N},[\frac{\pi}{6}+2k\pi,\frac{5\pi}{6}+2k\pi]$,$\frac{5\pi}{6}+2k\pi-(\frac{\pi}{6}+2k\pi)=\frac{4\pi}{6}>1$, hence there exists $n_k\in\mathbb{N},n_k\in[\frac{\pi}{6}+2k\pi,\frac{5\pi}{6}+2k\pi]$. Hence $|\sin n_k|>\frac{1}{2}$, and $\frac{1}{n_k}\geq \frac{1}{\frac{5\pi}{6}+2k\pi}>\frac{1}{(1+2k)\pi}$.

$\forall k\in\mathbb{N}$, we find such $n_k$, then $$\sum_{k=1}^\infty \frac{|\sin k|}{k}>\sum_{k=1}^\infty \frac{|\sin n_k|}{n_k}>\sum_{k=1}^\infty\frac{1}{(1+2k)2\pi}$$

the RHS series is divergent, by comparison test with hamonic series. So the series is not absolutely convergent.