Monday, 19 June 2017

sequences and series - Finding limntoinftyfracsinleft(fracpin+frac2pilog(n)sqrtnright)nlog(n)sinleft(fracpinright)



Is there a possible way to compute this limit ? I tried the L'Hôpital's rule, but it seems like the function become more complicated.



lim




All suggestions are welcomed.


Answer



First, \lim_n\frac{\sin\frac\pi n}{\frac1n}=\pi, so you might as well calculate
\lim_{n\to\infty} \dfrac{\sin(\frac{\pi}{n}+\frac{2 \pi \log(n)}{\sqrt n})}{\log n}.
You have
\left|\dfrac{\sin(\frac{\pi}{n}+\frac{2 \pi \log(n)}{\sqrt n})}{\log n}\right|\leq\frac1{\log n}\xrightarrow{\ \ \ \ \ }0,
so the limit is zero.


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