I know that:

$\displaystyle

c(n) = \sqrt{\frac{2}{n}}\ \frac{\Gamma(\frac{n}{2})}{\Gamma(\frac{n-1}{2})} =

1 - \frac{3}{4n} + O(n^{-2})

$

and also that:

$\displaystyle

\sqrt{\frac{n}{n-1}}\ c(n) = \sqrt{\frac{n}{n-1}}\ (1 - \frac{3}{4n} + O(n^{-2})) = (1 - \frac{1}{4n} + O(n^{-2}))

$

Problem is: I am not able to prove the very last equality. That's all what I need.

Seeking directions/help. Thanks a bunch!