Identity for central binomial coefficients On Wikipedia I came across the following equation for the central binomial coefficients:
$$
\binom{2n}{n}=\frac{4^n}{\sqrt{\pi n}}\left(1-\frac{c_n}{n}\right)
$$
for some $1/9<c_n<1/8$.
Does anyone know of a better reference for this fact than wikipedia or planet math? Also, does the equality continue to hold for positive real numbers $x$ instead of the integer $n$ if we replace the factorials involved in the definition of the binomial coefficient by Gamma functions?

It appears to be true for $x > .8305123339$ approximately: $c_x \to 0$ as $x \to 0+$. 
