Question: 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.

Answer: 
**Hint**

Since you want a direct comparison use this inequality

$$\sin(x)\le x,\; \forall x\ge0$$
It converges since it has positive terms and:
$$ \sin\dfrac1k \sim\_{\infty}\frac1k,\enspace\text{hence}\enspace\frac{ \sin\frac1k }{k}\sim\_{\infty}\frac1{k^2}.$$