# Converge? $\sum_{k=1}^{\infty}\frac{ \sin \left(\frac{1}{k}\right) }{k}$

Determine whether the series converges: $$\sum_{k=1}^{\infty}\frac{ \sin \left(\frac{1}{k}\right) }{k}$$

I tired to use direct comparison test but since 1/k is not convergent, so I am not sure about in this case.

$$\sin(x)\le x,\; \forall x\ge0$$
It converges absolutely: $\left|\sin\left(\dfrac{1}{k}\right)\right| < \dfrac{1}{k}$
It converges since it has positive terms and: $$\sin\dfrac1k \sim_{\infty}\frac1k,\enspace\text{hence}\enspace\frac{ \sin\frac1k }{k}\sim_{\infty}\frac1{k^2}.$$