Value of $\sum_{n=0}^{\infty}\binom{-1/2}{n}$

Question

I know that $\sum_{n=0}^{\infty}\binom{-1/2}{n}=2^{-\frac{1}{2}}$ from the Taylor series $(1+x)^p=\sum_{n=0}^{\infty}\binom{p}{n}x^n$. However, the Taylor series only converges for $|x|\lt1$ and $x=1$ in this case. I have already proved that $\sum_{n=0}^{\infty}\binom{-1/2}{n}$ converges I only need to prove that it converges to $2^{-\frac{1}{2}}$.

Answer 1

Since it converges, Abel's theorem tells you that\begin{align}\sum_{n=0}^\infty\binom{-1/2}n&=\lim_{x\to1^-}\sum_{n=0}^\infty\binom{-1/2}nx^n\\&=\lim_{x\to1^-}(1+x)^{-1/2}\\&=2^{-1/2}.\end{align}