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can you do favor and help me in solving this question : I want to study the series : $$\sum_{n=1}^{\infty}{\frac{\sin(\cos(2n))}{n}}.$$

I tried to use Abel and Dirichlet's theorem, but it seems impossible. So, do you have any suggestions? should I use Abel's transformation? or is there another way?

Thanks for help.

amWhy
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daniel11
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    Look at this post: https://math.stackexchange.com/questions/2280300/does-the-series-sum-limits-n-1-infty-frac-sin-left-cos2n-rightn-con/2280315#2280315 – Rigel May 14 '17 at 16:20
  • as you see, they aren't the same, this time $\sum_{n=1}^{\infty}{\frac{sin(cos(2n))}{n}}$ converge, but why?! – daniel11 May 14 '17 at 16:27
  • Please show us (i.e. edit into your question post). E.g., show how you applied Abel and Dirichlet's theorem, and why you ended up feeling it was impossible to finish? – amWhy May 14 '17 at 16:28
  • Please show how you have already concluded the series converges, @daniel? And since you already know the series "converge", why ask the question? – amWhy May 14 '17 at 16:31
  • using dirichlet's theorem, we should prove that $\sum_{k=1}^{n}{sin(cos(2k))},$ how should I prove that? – daniel11 May 14 '17 at 16:32
  • I used Mathematica, I got a real number, that how I know it converge. But I need a proof if anyone can help me. – daniel11 May 14 '17 at 16:34
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    while searching in your website, I found an interesting question, I think it is related to my question. https://math.stackexchange.com/questions/2239269/does-the-series-sum-n-1-infty-frac-sin-cosnn-converge?rq=1 – daniel11 May 14 '17 at 17:01
  • Congrats. Do you see the steps you need to use that result for yours? – Juanito May 14 '17 at 17:02
  • It is related to this one:https://math.stackexchange.com/questions/2239269/does-the-series-sum-n-1-infty-frac-sin-cosnn-converge – Sungjin Kim May 14 '17 at 18:15
  • By the method provided by @stewbasic, the convergence of the series follows. – Sungjin Kim May 14 '17 at 18:17
  • I see that OP found the link after posting the question. It's perfectly possible that attempts by Abel's or Dirichlet's theorem didn't help much in the first place. This requires more knowledge than OP attempted, and OP is asking about those. So, I do not think that this is a bad question. Now, a link with the similar question and answers are provided, and the question should have been resolved. – Sungjin Kim May 14 '17 at 18:30

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