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Question

Evaluate: $$\underset{n\rightarrow \infty}{lim}\left[\dfrac{n!}{n^n}\right]^{1/n}$$


Solution

Consider the given limit.


$$K=\displaystyle\underset{n\to \infty }{\mathop{\lim }}\,{{\left( \dfrac{n!}{{{n}^{n}}} \right)}^{1/n}}$$

Take log on both sides.

$$ \ln K=\displaystyle\underset{n\to \infty }{\mathop{\lim }}\,\dfrac{1}{n}\ln \left( \dfrac{n!}{{{n}^{n}}} \right) $$

$$ \ln K=\displaystyle\underset{n\to \infty }{\mathop{\lim }}\,\dfrac{1}{n}\left[ \ln \left( \dfrac{1}{n}.\dfrac{2}{n}.\dfrac{3}{n}........\dfrac{n}{n} \right) \right] $$

$$ \ln K=\displaystyle\underset{n\to \infty }{\mathop{\lim }}\,\dfrac{1}{n}\sum\limits_{r=1}^{n}{\ln \left( \dfrac{r}{n} \right)} $$

$$ \ln K=\int\limits_{0}^{1}{\ln xdx} $$

$$ \ln K=\left[ x\ln x \right]_{0}^{1}-\left[ x \right]_{0}^{1} $$

$$ \ln K=0-0-1 $$

$$ \ln K=-1 $$

$$ K={{e}^{-1}} $$

$$ K=\dfrac{1}{e} $$

Hence, this is the required value of the limit.


Mathematics

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