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Does the series $$ \sum_{n=1}^\infty \frac{\sin(2^n)}{n} $$ converge?

Based on Dirichlet's test, series like $$ \sum_{n=1}^\infty \frac{\sin(a[n])}{n} $$ converges when $a[n]$ is an arithmetic sequence. Different from this case, where $2^n$ is a geometric sequence. Not clear if Dirichet's test is still applicable.

gen-ℤ ready to perish
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    This is indeed a very hard problem. See this question – reuns Aug 30 '17 at 23:06
  • As far as I understand it : It is not hard to define a sequence $c_n$ such that the convergence of $\sum_n c_n$ is equivalent to an unproven conjecture, or even to an undecidable problem. What is intriguing here is that $\sum_n \frac{\sin(2^n)}{n}$ is easy to define and doesn't seem directly related to a known problem (in contrary to things like $\sum_n \frac{n^{-a}}{\sin(n)}$ depending on the irrationality measure of $\pi$) – reuns Aug 30 '17 at 23:12

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