Summation of special binomial series

Question

$\lim _{n \rightarrow \infty} \sum_{r=1}^{n}\left(\begin{array}{c}2 r \\ r\end{array}\right)\left(\frac{1}{6}\right)^{r}$

I have a feeling that this will be solved using some standard series but I cannot get around which series to use.

Answer 1

Hint

$$ \sum_{r=0}^{\infty}\left(\begin{array}{c}2 r \\ r\end{array}\right)x^{r}=\frac{1}{\sqrt{1-4 x}}$$