Value and limit of trigonometry sequence?

Question

I have a series: $\sum _{n=1}^{\infty }\tan \frac{1}{n},$

so it's

$\tan (1) + \tan \frac{1}{2} + \tan \frac{1}{3} + \dots$

I have an explenation that the series has a positive components, because $0 < \frac{1}{n} \le$ 1.

I have lack of basic knowledge, could somebody explain, how we can say, for example $\tan (\frac{1}{20})> 0 ?$

How to calculate the value of $\tan(\frac{1}{20})$ without radians?

The other question is, how to say that $lim_{n\to\infty} \tan\frac{1}{n} = 0?$

Answer 1

Hint:$$\forall x\in(0,\frac{\pi}2), \quad \tan x>x$$

Answer 2

Observe that

$$\frac{\tan\frac1n}{\frac1n}=\frac{\sin\frac1n}{\frac1n}\cdot\frac1{\cos\frac1n}\xrightarrow[n\to\infty]{}1\cdot1=1$$

so now apply the limit-comparison test for positive series.