so, we recently started to learn about series, i feel like i'm confused a little and, i seem like i don't know how to start with this,i guess maybe it's either comparison test, or limit comparison test, can anyone explain to me how to solve it?
$$\sum_{n=1}^∞ n\tan\frac{1}{n}$$
The series diverges, because for all $a \in [0,\frac {\pi}2)$, $\tan a \ge a$.
Therefore $n\tan\frac 1 n \ge n * \frac 1 n$, and latter is $1$.
We know that $\sum 1$ diverges, therefore, $$\sum n\tan\frac 1 n$$ diverges.
As for what test it is, it is comparison test.