Given the Probability Generating Function for a non-negative, integer-valued, R.V. $X$ as:
$$ g_X(t)=\log\left(\frac 1 {1-qt}\right). $$ How do I compute its Probability Function, $P(X=k)$? A step-by-step answer would be much appreciated.
The answer in the book is $$ P(X=k)=\frac {(1-e^{-1})^k} k. $$
The generating function is defined by $$g_X(t)=\sum_k P(X=k)t^k.\tag1$$ Taking $t=1$ we have $$\log\left(\frac1{1-q}\right)=g_X(1)=\sum_{k}P(X=k)=1$$ so we deduce that $-\log(1-q)=1$ and thus that $q=1-\mathrm e^{-1}$. To conclude, write the Taylor expansion in $t$ of $g_X$ $$g_X(t)=\log\left(\frac1{1-qt}\right)=-\log(1-qt)=\sum_{k=1}^\infty \frac{(qt)^k}k$$ by identification with (1), you get $P(X=k)=q^k/k$ as the answer says.