Divergence of series sin(1/n)
For $0\le x\le \pi/2$, the sine function is bounded by
$$\frac2\pi x\le \sin(x)\le x$$
Therefore, we assert that
$$\sum_{n=1}^N \sin\left(\frac1n\right) \ge \frac2\pi \sum_{n=1}^N \frac1n$$
And hence, by comparison with the harmonic series, the series $\sum_{n=1}^\infty \sin\left(\frac1n\right)$ diverges.
This is not valid. Indeed, note that
$$\lim_{n\to\infty}\sin(1/n)=0$$
So it passes the $n$th term test. However, it does diverge. Note that:
$$n\ge1\implies\sin(1/n)\ge\frac{\sin(1)}n$$
So we may use the direct comparison test.