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Find $a$ such that the limit is zero
Problem :
Let $x>0$ then define :
$$f(x)=\left(\left(\frac{1}{x}\right)!\left(x!\right)\right)^{\frac{1}{x+\frac{1}{x}}}$$
Then find $a$ such that :
$$\lim_{x\to\infty}f(x)-\frac{1}{2}\left(\frac{1}...
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Closed from for the series involving gamma function
Is there a closed form for the fallowing series,
$$\sum_{n=1}^\infty \Gamma\left(n+\frac12+\frac12k\right)\Gamma\left(n+\frac12-\frac12k\right)\frac{x^{n}}{(2n+1)!}$$
where $k\notin\mathbb{Z}$.
I ...
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