I have the following expression:
\begin{align} f &= \sum_{x=1}^\infty \int_{a - b^x}^{a - b^{x+1}} dF(k) \\ f &= \sum_{x=1}^\infty (F(a - b^{x+1}) - F(a - b^{x}) \end{align}
where $F$ is the CDF of a random variable, $b \in (0, 1)$, $a - b > 0$. I'm looking for a continuous random variable where $F(0) = 0$ under which this expression has an analytical solution that does not longer involve the sum. I'm not sure how to attack this problem, I've tried out a few known ones but they did not work out.
For example, many continuous random variables have a CDF involving the exponential, and equations of the form $\sum_{x=1}^\infty e^{a - b^x}$ do not permit a closed-form solution that does not involve the summation.
The sum here is a telescopic sum and its value is $F(a-)-F(a-b)$ for any CDF $F$.