Calculus (3rd Edition)

$\sum_{n=1}^{\infty}\sin(1/n)$ diverges.
By the limit comparison test, consider $b_n=\frac{1}{n}$. Now, we have $$L=\lim_{n\to \infty} \frac{\sin(1/n)}{1/n}=\lim_{1/n\to 0} \frac{\sin(1/n)}{1/n}=1.$$ Since $L\gt 1$ and the series $\sum_{n=1}^{\infty}\frac{1 }{n }$ is a divergent p-series, then $\sum_{n=1}^{\infty}\sin(1/n)$ diverges.