How to prove that this series diverges? I'm pretty sure I need to use the comparison test here, but I have no idea what to compare it to. I also know that it diverges only from the answers at the back of my book, how should I know to test it for divergence and not convergence?
$$\sum_{n=1}^\infty\sin\left(\frac{1}{n}\right)\cos\left(\frac{1}{n}\right)$$

$$\sin{\frac{1}{n}}\cos{\frac{1}{n}}$$
$$=\frac12 \sin{\frac{2}{n}}$$
$\text{Since } \sin(x) \ge \frac{2x}{\pi} \text{ for } 0 \le x \le \frac{\pi}{2}$:
$$\frac12 \sin{\frac{2}{n}} \gt \frac12 \cdot \frac{2}{\pi} \cdot \frac{2}{n} = \frac{2}{\pi} \cdot \frac{1}{n}$$
and the sum of this diverges.
