Given a function $G(x)= c \ln(1-{x \over 2})$ for $x \in [0,1]$ I first had to check for which $c$ the function $G$ is a probability generating function. I guess I need to check to property $G(1)=1$? If so, $G$ is a PGF for $c={1 \over \ln(0.5)}$.
Now for $X_c$ being the corresponding random variable, how can I determine the cumulative distribution function? I am not even sure what my random variable is, so I don't know how to make use of the definition $Pr(X_c \leq x)=...$
Some advice is much appreciated.
Recall that $$ \log(1-t) = -\sum_{k=1}^\infty \frac{x^k}k,\quad |t|<1. $$ Substituting $t=\frac12 x$ and writing $-\log\frac12 = \log 2$ we have $$ G(x) = \sum_{k=1}^\infty \frac1{\log(2) k2^k}x^k, $$ so that $$ \mathbb P(X_c=k) = \frac1{\log(2) k2^k}, $$ and hence $$ \mathbb P(X_c\leqslant x) = \sum_{k=1}^{\lfloor x\rfloor} \frac1{\log(2) k2^k}. $$