sequences and series - Study the convergence of $sum_nleft|frac{sin(alpha_n-alpha_m)}{alpha_n-alpha_m}right|$.

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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?

Answer

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.