Does the series $\displaystyle\sum_{k=1}^{\infty}\dfrac{k\sin(2kx)}{4k^2-1}$ converge? I tried to use the tests, but I couldn't get anything. Can anybody help me?
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1Welcome to MSE! Which tests have you tried? – mathcounterexamples.net Dec 23 '20 at 18:40
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1What do we know about $x$ ? – kmitov Dec 23 '20 at 18:42
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root and reason test. I couldn't find a function to try the comparison test. – Dec 23 '20 at 18:42
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@kmitov nothing.. $x \in \mathbb{R}$. – Dec 23 '20 at 18:43
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After the usual fiddling, this will reduce to this question. – user3482749 Dec 23 '20 at 18:43
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3I would suggest that you look at Dirichlet's test. – mathcounterexamples.net Dec 23 '20 at 18:44
1 Answers
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It converges, by Dirichlet's test, since:
- the sequence $\left(\frac k{4k^2-1}\right)_{k\in\Bbb N}$ is monotonic and converges to $0$;
- the series $\sum_{k=1}^\infty\sin(2kx)$ has bounded partial sums.
José Carlos Santos
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