Too many 5s

Algebra Level 4

If the sum \(\large \displaystyle \sum_{n=1}^{\infty} \dfrac{\overbrace{5555 \ldots 5}^{\text{n times}}}{13^n}\) can be represented as \(\dfrac{A}{B}\) where \(A\) and \(B\) are coprime positive integers, find \(A+B\).

Clarifications

The expanded form of above sum is \(\dfrac{5}{13} + \dfrac{55}{13^2} + \dfrac{555}{13^3} + \ldots\)

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