Does the series $\sum\limits_{n=1}^\infty \sin \dfrac{1}{n}$ converge or diverge?
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1Duplicate of Convergence/Divergence of $\sum_{n=1}^{\infty} \sin(1/n)$. – Andrés E. Caicedo Jul 17 '13 at 20:36
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1Comparison test and integral test don't appear to work for it – James Notari Jul 17 '13 at 20:37
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(re-opened and re-closed because of a typo in the question number of the previous closure target) – Willie Wong Jul 18 '13 at 07:22
2 Answers
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Since $$\lim_{n\to\infty}\frac{\sin\left(\frac{1}{n}\right)}{\frac{1}{n}}=1$$ so $$\sin\left(\frac{1}{n}\right)\sim_\infty \frac{1}{n}$$ so the series $\displaystyle\sum_{n\geq1} \sin\left(\frac{1}{n}\right)$ is divergent by comparison with the series $\displaystyle\sum_{n\geq1} \frac{1}{n}$.
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We have: $$ \forall x \in \left[0, \frac{\pi}{2}\right] : \frac{2x}{\pi} \le \sin x $$
Hence: $$ \frac{2}{\pi n} \le \sin\frac{1}{n} $$
Since $\sum 1/n$ diverges, $\sum \sin 1/n$ diverges too.
Ayman Hourieh
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