Sum of Series $ \frac{F_0}{6}+\frac{F_1}{6^1} +\frac{F_2}{6^2}+\frac{F_3}{6^3}+\frac{F_4}{6^4} + \ldots$ I want to find of the following series where $ F_0 = 1 $ , $ F_1 = 1 $ and $ F_n=F_{n-1}+F_{n -2}  $ .
$$  S = \frac{F_0}{6}+\frac{F_1}{6^1} +\frac{F_2}{6^2}+\frac{F_3}{6^3}+\frac{F_4}{6^4} + \ldots$$
I have tried to proceed with the thought of finding a close formula for this problem but cant be able to derive formula . Can you please help me to find the formula for this problem ? 

First, the series converges because $0 \leq F_n \leq 2^n$ by induction.
Then, you can compute
$$ S = \sum_{n=0}^{+\infty} \frac{F_n}{6^n}
     = F_0 + \frac{F_1}{6} + \sum_{n=2}^{+\infty} \frac{F_n}{6^n}
     = F_0 + \frac{F_1}{6} + \sum_{n=2}^{+\infty} \frac{F_{n-1}}{6^n} + \sum_{n=2}^{+\infty} \frac{F_{n-2}}{6^n}. $$
Reordering should get you to
$$ S = F_0 + \frac{F_1}{6} + \frac{S-F_0}{6} + \frac{S}{6^2}. $$
Then it's easy to find $S$.
