I am trying to determine whether the series $ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{|\sin(n)|}} $ converges or not. The difficulty is in that every now and then $\sin(n)$ will be very close to zero making the corresponding term in the series close to $\pm 1$. If I could show that these outlier terms eventually get much smaller than $1$, or if there are about as many positive such terms as negative ones then the series would converge, but I'm not sure how to approach that. Any help is appreciated.
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Are you familiar with the Denjoy-Koksma inequality? – Jack D'Aurizio May 04 '20 at 10:21
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https://mathoverflow.net/questions/282259/is-the-series-sum-n-sin-nn-n-convergent/282290#282290 – Alapan Das May 04 '20 at 10:23
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@Jack D'Aurizio Isn't that a bit heavy-handed? $\pi$ is irrational, so there are infinitely many $n$ with ${n/\pi}<1/n$, i.e. $|\sin(n)|<\pi/n$. – May 04 '20 at 19:09
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@ProfessorVector: you are right, that is enough. I was thinking to the more subtle case $$\sum_{n\geq 1}\frac{1}{n^{1+|\sin(n)|}}.$$ – Jack D'Aurizio May 05 '20 at 00:48
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I think I figured it out. By what @ProfessorVector said there will be infinitely many terms in the series satisfying $$ \frac{1}{n^{\frac{\pi}{n}}} < \frac{1}{n^{|\sin(n)|}} <1 $$ in magnitude. Since the left side of the inequality tends to 1 as $n$ tends to infinity, there will be infinitely many terms arbitrarily close to 1 in magnitude. Because of this the series diverges. – math_guy May 05 '20 at 14:52
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Thanks everybody! – math_guy May 05 '20 at 15:16