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Actually, I have tried in many ways, first of all for test of absolute convergence, but as the sequence $\{\cos{(n)}\}$ is dense in $[-1,1]$, hence many of the tests will not respond, and if $\{S(n)\}$ is the sequence of partial sum of the given series, then I tried to show that one of it's subsequences may diverge, as because I have tested some $S(n)$, but they are switching positive and off positive, hence I think it will diverge .

rtybase
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Rabi Kumar Chakraborty
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    Related: https://math.stackexchange.com/questions/2684857/does-sum-n-1-infty-sinnn-converge?noredirect=1&lq=1 – Robert Z Nov 03 '18 at 08:43
  • Not worth much but the series DOES converge to around 10.65 surprisingly: https://www.wolframalpha.com/input/?i=sum+from+n+%3D+1+to+infinity+of+(cos(n))%5En

    Not sure how to show to the convergence myself though

     – PrincessEev Nov 03 '18 at 08:55
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    @EeveeTrainer Just because WA gives a value doesn't mean it converges. It evaluates it numerically. If the growth is slow enough it can think it has converged without this being the case. Try for example this and compare with this answer. The fact that $\sum \sin(n)^n$ diverges probably means this series also diverges (the same argument can likely be extended to this case, but I haven't checked this). – Winther Nov 03 '18 at 09:00
  • Huh. My bad then, thanks for letting me know. – PrincessEev Nov 03 '18 at 09:03
  • Regardless if it converges, Jack would probably be the expert of explaining why. – mathreadler Nov 03 '18 at 11:27
  • There is some uniform sampling theorem for periodic functions involving an integral you can probably use. – mathreadler Nov 03 '18 at 11:28
  • Thanks, Robert Z , I have seen the answer, and the trick actually, I think it can be done only using the Dirichlet's irrationality criterion, in almost the same way, actually. – Rabi Kumar Chakraborty Nov 03 '18 at 11:30
  • Maybe testing whether the sequence is cauchy can help. – DreaDk Nov 03 '18 at 11:43
  • See also https://math.stackexchange.com/questions/2898385/when-does-the-limit-cosnfn-converges-as-n-rightarrow-infty-n-in/2901343#2901343 – Sungjin Kim Nov 04 '18 at 17:15

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