Sunday, August 18, 2019

Convergence of the series sum+inftyn=1left(nsinfrac1nright)n



I have to study the convergence of the series



+n=1(nsin1n)n



and



+n=1((nsin1n)n1).



I know I should study the limit




lim



and that



\lim_{n\to +\infty}{n\sin{\frac{1}{n}}} = 1




but I don't see how it helps. Any ideas ?



Thank you in advance !


Answer



On the interval (0,1) we have
1-\frac{x^2}{6} \leq \frac{\sin x}{x}\leq e^{-x^2/6}
hence \left(n\sin\frac{1}{n}\right)^n behaves like e^{-\frac{1}{6n}}=1-\frac{1}{6n}+O\left(\frac{1}{n^2}\right) for large values of n and the given series are divergent.


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