Fibonacci on fives

Calculus Level 4

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

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.

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