Inspired by today's post in MSE:
A sum of absolute values of binomial coefficients
my numerical experiments at Mathematica throw up: $$2^{-x}\sum_{k=0}^{\infty} {x \choose k} =1, x\in R^{+},$$ fairly accurately. When $x$ is positive integer it is the well known identity. However for real positive values it is surprising ! Even if $x$ is non-real such that $\Re(x) >0$, it seems to hold fairly accurately. Any comment, Reference of proof is welcome.