The problem is from Varadhan's Probability Theory, p.39, EXERCISE 3.7.
Can you calculate the geometric ratio $$\rho(x)=\lim_{n\rightarrow \infty}\left(\sum_{r\geq nx} \binom{n}{r}\frac{1}{2^n}\right)^{1/n}$$ explicitly as a function of $x$ for $x>1/2$?