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When does $\sum_{n=1}^{\infty}\frac{\sin n}{n^p}$ absolutely converge?

Does $\sum_{n=1}^{\infty} \frac{\sin(n)}{n}$ converge conditionally?

How to prove that $ \sum_{n \in \mathbb{N} } | \frac{\sin( n)}{n} | $ diverges?

(for example) is studied the convergence of the series $$\sum_n\left|\frac{\sin n}{n}\right|$$ In this question let us consider a real sequence $\alpha_n$. About this sequence there are no hypotheses, except that $$|\alpha_n-\alpha_{n-1}|\geq \gamma>0$$ Study the convergence of $$\sum_n\left|\frac{\sin(\alpha_n-\alpha_m)}{\alpha_n-\alpha_m}\right|$$

I´m not able to verify it. Any suggestions please?

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Mark
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Suppose $a_n = \pi n/2$. Then $|a_n-a_m| =\pi|n-m|/2 $ and $|\sin(a_n-a_m)| =|\sin(\pi(n-m)/2)| =1 $ whenever $n-m$ is odd and zero otherwise.

Therefore $S(m) =\sum_{n \ne m} \big| \frac{\sin(a_n-a_m)}{a_n-a_m} \big| $ diverges for all $m$ by comparison with the harmonic series.

marty cohen
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