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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 \[\sum_{n=1}^{\infty}\frac{F_n}{3^n}=\frac{1}{3} + \frac{1}{9} + \frac{2}{27} + \cdots + \frac{F_n}{3^n} + \cdots.\]

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