Fibonacci on fives

Calculus Level 4

S=15+152+253+354+555+856+ 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=ABS = \dfrac AB, where AA and BB are coprime positive integers, find the value of A+BA+B.

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

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