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Are these series convergent or divergent?

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

and

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

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2 Answers

up vote 4 down vote accepted

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.

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