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Evaluating a power series

Can somone help me find a closed form expression for this sum given any rational value of x, and any integer p, where {x} denotates the fractional part of x.$$\sum_ {k=1}^{\infty}\frac{ \left\{p^kx \right\} }{p^k} $$ It also seems to converge to rational values.

For example if i let $p=5,x=\frac13$, the series converges to $\frac{11}{72}$

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