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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!

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Hint: For $x \in (0,\tfrac{\pi}{2})$ we have $\tan x > x$. Thus, $\tan \dfrac{1}{n} > \dfrac{1}{n}$. Now use the comparison test.

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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.

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