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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