The question is to find the moment generating function of
p(k) = Pr(X = k) = −p^k/(k*ln(1-p) where p<1 and k=1,2,3...
Putting this into the formula, I have
Σe^(t*k)p^k/(xlog(1-p))
but I am not sure what to do next.
Any help would be much appreciated!
Assuming that the PMF is - $$P\left(X=k\right) =\frac{-p^k}{k.\ln \left(1-p\right)}$$ The MGF is given by - $$∑_{k=1}^∞\frac{-p^ke^{tk}}{\left(k.\ln \left(1-p\right)\right)}$$
$$ = ∑_{k=1}^∞\frac{{-\left(pe^{t}\right)}^{k}}{\left(k.\ln \left(1-p\right)\right)}$$ $$ = \frac{1}{\ln \left(1-p\right)}∑_{k=1}^∞\frac{{-\left(pe^{t}\right)}^{k}}{k}$$ $$ = \frac{\ln \left(1-pe^t\right)}{\ln \left(1-p\right)}$$