calculus - Does the series $sumlimits_{n=1}^{infty}ntanleft(frac {pi}{2^{n+1}}right)$ converge or diverge?

Does the series $\sum\limits_{n=1}^{ \infty}n\tan\left( \dfrac { \pi}{2^{n+1}}\right)$ converge or diverge? My idea was to use the limit comparison test and $\sum\limits_{n=1}^{\infty} \dfrac {n}{2^{n}}$, but then I don't know what to do with the tangent which in the limit is 0.

Answer

Hint: note that
$$
\lim_{n \to \infty} \frac{\tan\left(\frac{\pi}{2^{n+1}}\right)}{\left(\frac{\pi}{2^{n+1}}\right)} = 1
$$