A Diminishing Geometric Distribution

A standard geometric distribution can be interpreted as the number of Bernoulli trials required to get one success. However, what if the probability of success if each trial diminishes by some factor with each failure?

Let p be probability of success of the first trial and d be the diminishing factor of each failure such that the probability of success of trial n is $p{d}^{n-1}$. I have been trying to calculate the expected value of this distribution without success. My main stumbling block is the probability of failing n times:

$P(n\text{failures})=\prod _{k=1}^{n}(1-p{d}^{k-1})$

Given that, the expected value would be

$E(X)=\sum _{n=1}^{\mathrm{\infty}}np{d}^{n-1}F(n-1)$

Is there a way to get a closed form for this?

A standard geometric distribution can be interpreted as the number of Bernoulli trials required to get one success. However, what if the probability of success if each trial diminishes by some factor with each failure?

Let p be probability of success of the first trial and d be the diminishing factor of each failure such that the probability of success of trial n is $p{d}^{n-1}$. I have been trying to calculate the expected value of this distribution without success. My main stumbling block is the probability of failing n times:

$P(n\text{failures})=\prod _{k=1}^{n}(1-p{d}^{k-1})$

Given that, the expected value would be

$E(X)=\sum _{n=1}^{\mathrm{\infty}}np{d}^{n-1}F(n-1)$

Is there a way to get a closed form for this?