Thursday, 29 March 2018

limits - Find limnrightarrowinftyfracn(n!)1/n





I am having trouble showing lim


Answer



Let a_{n}=n^{n}/n! and then a_{n+1}/a_{n}=\{(n +1)^{n+1}/(n+1)!\}\{n!/n^{n}\}=\left(1+\frac{1}{n}\right)^{n} which tends to e and therefore a_{n}^{1/n} also tends to e.


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