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Determine whether the series converges: $$\sum_{k=1}^{\infty}\frac{ \sin \left(\frac{1}{k}\right) }{k} $$

I tired to use direct comparison test but since 1/k is not convergent, so I am not sure about in this case.

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3 Answers 3

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Hint

Since you want a direct comparison use this inequality

$$\sin(x)\le x,\; \forall x\ge0$$

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It converges absolutely: $\left|\sin\left(\dfrac{1}{k}\right)\right| < \dfrac{1}{k}$

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It converges since it has positive terms and: $$ \sin\dfrac1k \sim_{\infty}\frac1k,\enspace\text{hence}\enspace\frac{ \sin\frac1k }{k}\sim_{\infty}\frac1{k^2}.$$

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  • $\begingroup$ It's useless to say that it converges absolutely since it's a positive series. $\endgroup$
    – user296113
    Mar 15, 2016 at 18:33
  • $\begingroup$ An old habit. It may happen the first terms of a series are not positive., even if they're eventually positive. $\endgroup$
    – Bernard
    Mar 15, 2016 at 18:36

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