Friday, July 6, 2018

limits - Compute limnrightarrowinftyfracnn(n!)2



I have to compute limnnn(n!)2.



I tried say that this limit exists and it's l, so we have limnnn(n!)2=L then I rewrited it as:
limn(nnn!)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



(n+1)n+1((n+1)!)2(n!)2nn=1n+1(1+1n)n0



then



limnnn(n!)2=0


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