Fibonacci on fives

Calculus Level 3

$S =\frac{1}{5} + \frac{1}{5^{2}} + \frac{2}{5^{3}} + \frac{3}{5^{4}} + \frac{5}{5^{5}} + \frac{8}{5^{6}} + \cdots$

If $S = \dfrac AB$ , where $A$ and $B$ are coprime positive integers, find the value of $A+B$ .

Clarification: $1,1,2,3,5,8,\ldots$ follows the Fibonacci Sequence.