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Determine whether the sum converges $\sum\limits_{n=1}^{\infty} \frac{\sin^n(n)}{n} $

I tried to use all the standard signs, but there's nothing to understand about them in general. It seems to me that the problem is connected with the approximation of the number pi by rational numbers, but I could not bring this idea to the end.

Gabriel Romon
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Vladislav Kharlamov
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    Related: https://math.stackexchange.com/questions/823816/is-sum-limits-n-1-infty-frac-sin-nnn-convergent and https://math.stackexchange.com/questions/811717/does-the-sequence-sinnn-converge/ – Gabriel Romon May 08 '18 at 20:43
  • Thanks you!.... – Vladislav Kharlamov May 08 '18 at 20:46
  • In fact, the question is a duplicate of https://math.stackexchange.com/questions/328306/convergence-of-the-series-sum-n-1-infty-frac-sin-nnn . But since the latter has been left unanswered, I don't think yours should be closed. – Gabriel Romon May 08 '18 at 20:48

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This series converges, as do a variety of similar series. This follows from a mild generalization of a paper by Ravi B. Boppana which was about a similar problem. Just a few months ago Ravi uploaded a YouTube video detailing his proof. The video is very informative and not too long, so I definitely recommend it. Over at this duplicate question I posted a full detailed proof of a generalized fact, which answers this question directly. I proved that for all real numbers $\alpha\in (0,1]$, the following series is absolutely convergent, where the title question is the special case $\alpha=1$. $$\sum_{n=1}^\infty \frac{((1-\alpha)+\alpha\sin(n))^n}{n}$$

Jade Vanadium
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