How can I show binomial series converges to $\sqrt {2}$?

Question

$\sum_{n=0}^{\infty} 2n\binom{-\frac{1}{2}}{n}(-\frac{1}{2})^n = \sqrt{2}$

From wolfram alpha, it says that above series including binomial term $\binom{-\frac{1}{2}}{n}$ converges to $\sqrt{2}$.

I tired to convert it to $(1+x)^{-\frac{1}{2}}$ but I can't because of $2n$.

How can I show this binomial series converges to $\sqrt 2$ ?

Answer 1

For $\lvert x\rvert<1$,

$$\sum_{n=0}^\infty 2n\binom{-1/2}{n}x^n=2x\sum_{n=0}^\infty n\binom{-1/2}{n}x^{n-1}=2x\frac d{dx}\left[\sum_{n=0}^\infty\binom{-1/2}n x^n\right]=-x(x+1)^{-3/2}$$

Hence the result for $x=-\frac12$.