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Question Number 188826 by mathlove last updated on 08/Mar/23

prove that  lim_(n→∞) (((1! 2! 3!∙∙∙∙∙n!))^(1/(n(n+1))) /( (√n)))=e^((−3)/4)

$${prove}\:{that} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\sqrt[{{n}\left({n}+\mathrm{1}\right)}]{\mathrm{1}!\:\mathrm{2}!\:\mathrm{3}!\centerdot\centerdot\centerdot\centerdot\centerdot{n}!}}{\:\sqrt{{n}}}={e}^{\frac{−\mathrm{3}}{\mathrm{4}}} \\ $$

Commented by Frix last updated on 07/Mar/23

There is no x and why →0?

$$\mathrm{There}\:\mathrm{is}\:\mathrm{no}\:{x}\:\mathrm{and}\:\mathrm{why}\:\rightarrow\mathrm{0}? \\ $$

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