Friday, July 6, 2018

limits - Compute limnrightarrowinftyfracnn(n!)2



I have to compute lim.



I tried say that this limit exists and it's l, so we have \lim_{n\rightarrow\infty}\frac{n^n}{(n!)^2} = L then I rewrited it as:
\lim_{n\rightarrow\infty}(\frac{\sqrt n}{\sqrt[n]{n!}})^{2n} then I used natural log over the whole expresion but didn't got into a nice place.




I don't know about Pi function or gamma function so therefore can't really use L'Hospital's rule.


Answer



By ratio test



\frac{(n+1)^{n+1}}{((n+1)!)^2}\frac{(n!)^2}{n^n}=\frac1{n+1}\left(1+\frac1n\right)^n\to 0



then



\lim_{n\rightarrow\infty}\frac{n^n}{(n!)^2}=0


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