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For this problem, I am tasked with finding the moment generating function of $X$ given it has the following discrete probability distribution: $$f(x)=2*(\frac{1}{3})^x, x\in\mathbb{N}$$ By the definition of the moment generating function, $M_{xt}=\mathbb{E}(e^{xt})$. In this case, the expectation would be equal to: $$\sum_{x=1}^\infty 2*(\frac{1}{3})^x e^{xt}=2\sum_{x=1}^\infty(\frac{1}{3})^x e^{xt}=2\sum_{x=1}^\infty(\frac{e^t}{3})^x$$ I am frankly unsure what to do with the last sum. Indeed, it seems the sum will not even converge for $t\geq$ln$3$. Some help getting past this roadblock would be appreciated.

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The sum equals $$ \frac{2\exp(t)/3}{1-(\exp(t)/3)}=\frac{2e^t}{3-e^t} $$ for $t<\log 3$. This is not an issue because there is not a guarantee that (if $X\geq 0$), that $Ee^{tX}<\infty$ for all $t$. In particular $Ee^{5X}=\infty$ in this case.

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