TITLE: $\sum \tan ( 1/n)$ diverges
QUESTION [8 upvotes]: Show that the series
$$\sum_n \tan\left(\frac{1}{n}\right)$$
diverges.
I dont have any attempt to do, since I am having some troubles with series including geometric functions. I would be glad if I could get a detailed answer, if it is possible.
Thanks!

REPLY [8 votes]: As $x \to 0$, we have that $\frac{\tan x}{x} \to 1$. So $\tan \frac{1}{n}$ behaves just like $\frac 1n$ as $n$ gets large.