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By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent.

  • Is is true that $$\frac{1}{2}=\inf\left\{\alpha\in\mathbb{R}^+:\sum_{n\geq 1}\frac{\sin(n^2)}{n^\alpha}\mbox{ is convergent}\right\}?$$
  • In the case of a positive answer to the previous question, what is $$\inf\left\{\beta\in\mathbb{R}^+:\sum_{n\geq 1}\frac{\sin(n^2)}{\sqrt{n}(\log n)^\beta}\mbox{ is convergent}\right\}?$$
  • 2
    By modelling $\sin(n^2)$ as a sequence of independent random variables $X_n$, I would expect a positive answer to the first question, also I would expect the later series to be convergent when $\beta > 1/2$ and divergent when $\beta < 1/2$. Thus the answer to the second question would be $1/2$.2012-11-30
  • 0
    [A related question](http://math.stackexchange.com/questions/335104/prove-that-sum-k-0neik2-on-alpha-forall-alpha-0/342356#342356) has popped up since this one was asked.2013-03-29
  • 1
    Applying the argument here, http://math.stackexchange.com/questions/2270/convergence-of-sum-limits-n-1-infty-sinnk-n/2275#2275 we obtain that the first quantity is $\leq \frac{7}{8}$2013-06-08

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