Combinatorics involving negative rational n in nCr function

Show that $\binom{-1/2}{k} $ =$(\frac{-1}{4}) ^{k} \times \binom{2k}{k} $ 

Hence, show that
$\frac{1}{\sqrt{1+x} }=\sum_{k=0}^{\infty }\binom{2k}{k}(\frac{-x}{4})^{k} $

  • Knowledge of the gamma function is not assumed for this question.

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