AoPS Wiki:Problem of the Day/September 10, 2011

Let $F_0 = 0$ , $F_1 = 1$ , and $F_n = F_{n - 1} + F_{n - 2}$ . Find the value of the infinite sum $\sum_{n=1}^{\infty}\frac{F_n}{3^n}=\frac{1}{3} + \frac{1}{9} + \frac{2}{27} + \cdots + \frac{F_n}{3^n} + \cdots.$