Are these series convergent or divergent?
$ \sum_{}^\infty [\sin(\frac{n\pi}{6})]^n $
and
$\sum_{}^\infty [\sin(\frac{n\pi}{7})]^n $
Are these series convergent or divergent?
$ \sum_{}^\infty [\sin(\frac{n\pi}{6})]^n $
and
$\sum_{}^\infty [\sin(\frac{n\pi}{7})]^n $
The values of the sine cycle through a finite number of values. You can easily see that $\limsup_{n\rightarrow\infty} \left|\sin\left(\frac{n\pi}{6}\right)\right| = 1$ while $\limsup_{n\rightarrow\infty} \left|\sin\left(\frac{n\pi}{7}\right)\right| < 1$ This means the first sum cannot converge while the second sum will be absolutely convergent.
Taking $\,n=3k\,\,,\,k\in\Bbb N\,\,,\,\,k\,\,\text{odd}$ , we get
$\sin\frac{n\pi}{6}=\sin\frac{k\pi}{2}=\pm 1\Longrightarrow \sin^n\frac{n\pi}{2}\rlap{\;\;\;\;\;/}\xrightarrow [n\to\infty]{}0$
so the series cannot converge.