lim_(x→∞) ((((x!)/x^x ))^(1/x) )^(((2∙4)/(1∙3))∙((4∙6)/(3∙5))∙∙∙) =? (2/1)∙(2/3)∙(4/3)∙(4/5)∙(6/5)∙(6/7)∙∙∙=(π/2) lim_(x→∞) ((((x!)/x^x ))^(1/x) )^((2/1)∙(4/3)∙(4/3)∙(6/5)∙∙∙) =((1/e))^(π/2) =(√(((1/e))^π ))=(1/( (√e^π )))


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Question Number 118967 by obaidullah last updated on 21/Oct/20

$\lim_{x \to \infty} {({(\frac{x!}{x^{x}})}^{\frac{1}{x}})}^{\frac{2 \cdot 4}{1 \cdot 3} \cdot \frac{4 \cdot 6}{3 \cdot 5} \cdot \cdot \cdot} = ?$ $\frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \cdot \cdot = \frac{π}{2}$ $\lim_{x \to \infty} {({(\frac{x!}{x^{x}})}^{\frac{1}{x}})}^{\frac{2}{1} \cdot \frac{4}{3} \cdot \frac{4}{3} \cdot \frac{6}{5} \cdot \cdot \cdot} = {(\frac{1}{e})}^{\frac{π}{2}} = \sqrt{{(\frac{1}{e})}^{π}} = \frac{1}{\sqrt{e^{π}}}$
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