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Are these series convergent or divergent?

$ \sum_{}^\infty [\sin(\frac{n\pi}{6})]^n $

and

$\sum_{}^\infty [\sin(\frac{n\pi}{7})]^n $

2 Answers 2

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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.

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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.