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.
[Math] Does the series $\sum\limits_{n=1}^{\infty}n\tan\left(\frac {\pi}{2^{n+1}}\right)$ converge or diverge
calculusconvergence-divergencelimitssequences-and-series
Best Answer
Hint: note that $$ \lim_{n \to \infty} \frac{\tan\left(\frac{\pi}{2^{n+1}}\right)}{\left(\frac{\pi}{2^{n+1}}\right)} = 1 $$