Find the moment generating function of the discrete random variable X that has the probability distribution
f(x) = 2($\frac13)^x$ for x=1,2,3...
$$ M_x(t) = \sum_x e^{tx}\cdot 2(\frac13)^x $$
$$ M_x(t) = 2\sum_xe^t\cdot e^x \cdot (\frac13)^x $$
$$ M_x(t) = 2e^t \sum_x (e/3)^x $$
I have no clue where to go from here. Can I get some help? I think it has to do with a geometric series
The problem is that $e^{tx}\neq e^{t}e^x$ in general. However, since $$e^{tx}\cdot 2\left(\frac13\right)^x=2\left(\frac{e^t}3\right)^x,$$ the sum giving $M_X$ can be computed like a geometric series.