Find Moment Generating Function from Probability Mass Function

Question

I need help understanding how to find the MGF using a PMF. The PMF is $f(x) = \frac{1}{2^{x-1}}$ when the random variable $X \geq 2$. I get that you need to multiply $e^{tx}$ by $\frac{1}{2^{x-1}}$. But I don't know where to go from there.

Answer 1

$$\Bbb E[e^{tX}] = \sum_{k=2}^\infty \frac{e^{tk}}{2^{k-1}} = e^t\sum_{k=2}^\infty\frac{e^{t(k-1)}}{2^{k-1}} = e^t\sum_{k=1}^\infty \left(\frac{e^t}{2}\right)^k$$ For $e^t < 2$, this is a geometric series missing the $0^{th}$ term, so: $$\Bbb E[e^{tX}] = e^t\left(\frac{1}{1-\frac{e^t}{2}} -1\right) = e^t \left(\frac{e^t}{2-e^t}\right) = \frac{e^{2t}}{2-e^t}$$ valid on $t < \log(2)$.

Answer 2

$M_{X}(t) = \mathbb{E}[e^{tX}]$

$\textbf{Discrete case:}$ $\mathbb{E}[e^{tX}] = \sum e^{tx}p(x)$

$\textbf{Continuous case:}$ $\mathbb{E}[e^{tX}] = \displaystyle \int e^{tx}p(x)\mathrm{d}x$