Prove that
$\displaystyle \sum_{n=0}^{\infty}{{2n}\choose n} x^n$ diverges for $\displaystyle |x| = 1/4$.
Also find the closed-form of the series for x in the radius of convergence.
For $\displaystyle x=1/4$ I think you can use Abel's theorem. It would mean the series is continous at $\displaystyle x=1/4$ which is a problem because it is equal to $\displaystyle (1-4x)^{-1/2}$ when $\displaystyle |x|<1/4$ and therefore not defined at $\displaystyle x=1/4$.