Answer
Diverges
Work Step by Step
We know that $\tan\frac{1}{n}\leq \tan 1\approx 1.56$ for all $n\geq 1$ since for $0\leq x\leq 1$ the function $\tan x$ is increasing.
Then,
$\frac{1}{n}\tan\frac{1}{n}\geq \frac{1}{n}$ for all $n$
Since the series $\sum_{n=1}^\infty\frac{1}{n}$ diverges, it follows by the Direct Comparison Test with $a_n=\frac{1}{n}\tan\frac{1}{n}$ and $b_n=\frac{1}{n}$ that the series $\sum_{n=1}^\infty\frac{1}{n}\tan\frac{1}{n}$ diverges.
