Question

نشان دهید که: $$ \lim_{x\to 0} \frac{ \sqrt[x!-1]{( \frac{(x+2)!}{2})! } }{(x-1)!sinx} = e^{ \frac{( \gamma -1)(3-2 \gamma )}{2 \gamma } } $$

Comments

$$x!= \Gamma (x+1) \wedge ( \frac{(x+2)!}{2} +1)!= \Gamma ( \frac{ \Gamma (x+3)}{2} +1)$$$$ \lim_{x\to 0}  \frac{ \sqrt[ \Gamma (x+1)-1]{ \Gamma [ \frac{ \Gamma (x+3)}{2} +1]} }{x(x-1)! \frac{sinx}{x} } = \lim_{x\to 0} [ \Gamma ( \frac{ \Gamma (x+3)}{2} +1)]^{ \frac{1}{ \Gamma (x+1)-1} }=exp[ \lim_{x\to 0}  \frac{log[ \Gamma ( \frac{ \Gamma (x+3)}{2} +1)]}{ \Gamma (x+1)-1}  $$$$=exp[ \lim_{x\to 0}  \frac{ \frac{ \frac{d}{dx}[ \Gamma ( \frac{ \Gamma (x+3)}{2} +1)] }{[ \Gamma ( \frac{ \Gamma (x+3)}{2} +1)]} }{ \Gamma '(x+1)} ]=exp[ \lim_{x\to 0}  \frac{ \Gamma '[ \frac{ \Gamma (x+3)}{2} +1] \frac{ \Gamma '(x+3)}{2} }{ \Gamma '(x+1)[ \Gamma ( \frac{ \Gamma (x+3)}{2} +1)]} ] \wedge  \frac{d}{dx} ln \Gamma (x)= \frac{ \Gamma '(x)}{ \Gamma (x)}= \psi (x) $$$$ \psi (n)=- \gamma +H_ {n-1} \Longrightarrow exp[ \frac{ \Gamma '(2) \frac{ \Gamma '(3)}{2} }{ \Gamma (2) \Gamma '(1)} ] =exp[ \frac{1}{2}  \frac{ \Gamma (3)  \psi  (3) \Gamma (2)  \psi  (2)}{ \Gamma (1) \psi (1)} ]$$

Answer 1

$$x!= \Gamma (x+1) \wedge ( \frac{(x+2)!}{2} +1)!= \Gamma ( \frac{ \Gamma (x+3)}{2} +1)$$$$ \lim_{x\to 0} \frac{ \sqrt[ \Gamma (x+1)-1]{ \Gamma [ \frac{ \Gamma (x+3)}{2} +1]} }{x(x-1)! \frac{sinx}{x} } = \lim_{x\to 0} [ \Gamma ( \frac{ \Gamma (x+3)}{2} +1)]^{ \frac{1}{ \Gamma (x+1)-1} }=exp[ \lim_{x\to 0} \frac{log[ \Gamma ( \frac{ \Gamma (x+3)}{2} +1)]}{ \Gamma (x+1)-1} $$$$=exp[ \lim_{x\to 0} \frac{ \frac{ \frac{d}{dx}[ \Gamma ( \frac{ \Gamma (x+3)}{2} +1)] }{[ \Gamma ( \frac{ \Gamma (x+3)}{2} +1)]} }{ \Gamma '(x+1)} ]=exp[ \lim_{x\to 0} \frac{ \frac{d}{dx} [ \Gamma ( \frac{ \Gamma (x+3)}{2} +1)}{[ \Gamma ( \frac{ \Gamma (x+3)}{2} +1)] \Gamma '(x+1)} ]=exp[ \lim_{x\to 0} \frac{ \Gamma '[ \frac{ \Gamma (x+3)}{2} +1] \frac{ \Gamma '(x+3)}{2} }{ \Gamma '(x+1)[ \Gamma ( \frac{ \Gamma (x+3)}{2} +1)]} ] \wedge \frac{d}{dx} ln \Gamma (x)= \frac{ \Gamma '(x)}{ \Gamma (x)}= \psi (x) $$$$ \psi (n)=- \gamma +H_ {n-1} \Longrightarrow exp[ \frac{ \Gamma '(2) \frac{ \Gamma '(3)}{2} }{ \Gamma (2) \Gamma '(1)} ] =exp[ \frac{1}{2} \frac{ \Gamma (3) \psi (3) \Gamma (2) \psi (2)}{ \Gamma (1) \psi (1)} ] \wedge \psi (1)=- \gamma \wedge \psi (2)=- \gamma +H_1=- \gamma +1 \wedge \psi (3)=- \gamma +H_2=- \gamma +1+ \frac{1}{2} =- \gamma + \frac{3}{2} \Longrightarrow ?=exp[ \frac{1}{2} \frac{2(- \gamma + \frac{3}{2})(- \gamma +1) }{- \gamma }]=exp[ \frac{(\gamma-1)(3-2 \gamma ) }{2 \gamma } ]$$