Probability Generating Function Attempt

Question

I am trying to find the PGF for the following distribution:

$X_1$ has PMF: $\rho(x) = \frac{-p^x}{x\ln(1-p)},n\in \mathbb N$

Attempt: \begin{align*} \phi_{X}(s) &= E[s^x]\\ &=\sum^{\infty}_{x=1} s^x \frac{-p^x}{x\ln(1-p)}\\ &=\frac{1}{\ln(1-p)} \sum^{\infty}_{x=1} \frac{-(sp)^x}{x}\frac{\ln(1-sp)}{\ln(1-sp)}\\ &=\frac{\ln(1-sp)}{\ln(1-p)}, ~~~~~~~~~s<\frac{1}{p} \end{align*}

Just found that this is a well known distribution called the log series distribution. My result is correct as verified by:

http://mathworld.wolfram.com/Log-SeriesDistribution.html

Answer 1

Hint: you already did a very similar sum in showing that your distribution actually is a probability distribution.