Is the sum $$\sum_{n=1}^\infty \frac{\sqrt{n} - \sin \frac{1}{\sqrt{n}}}{n}$$ convergent?
Convergence of $\sum \frac{\sqrt{n} - \sin \frac{1}{\sqrt{n}}}{n}$.
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calculus
sequences-and-series
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0what does $\sqrt{-}$ mean? – 2017-02-03
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0I'd like to edit it, but cannot. It should be $\sqrt{n} - $. The correct formula is $\sum \frac{\sqrt{n}-\sin{\frac{1}{\sqrt{n}}{n}$. – 2017-02-03
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0why can't you edit it? I can do it for you – 2017-02-03
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0Have no idea..it's correct now, thanks! – 2017-02-03
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0Divergent: Comparison test. – 2017-02-03
1 Answers
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Hint $$\frac{\sqrt{n} - \sin \frac{1}{\sqrt{n}}}{n}\geq\frac{\sqrt{n} - \frac{1}{\sqrt{n}}}{n}=\frac{n-1}{n\sqrt{n}}\geq\frac{n-\frac{n}{2}}{n\sqrt{n}}=\frac{1}{2\sqrt{n}}$$