Recursion, Catalan Numbers & Generating Functions · Challenge problem

The Catalan numbers are $C_0=1$, $C_1=1$, $C_2=2$, $C_3=5$, $C_4=14,\dots$, with $C_n=\dfrac{1}{n+1}\dbinom{2n}{n}$. Evaluate $\displaystyle\sum_{n=0}^{\infty}C_n\left(\dfrac29\right)^{n}$. Express your answer as a common fraction.