Why does the moment generating function of a chi-squared random variable only exist for $t<1/2$? I have found that for a chi-squared ($n$ degrees of freedom) random variable $X$, $M_X(t)= (1-2t)^{-n/2}$. I am told that this only exists for $t<1/2$.  Why is this? 
 A: For $X \sim \chi_n^2$ we have moment generating function:
$$\begin{equation} \begin{aligned}
M_X(t) \equiv \mathbb{E}(\exp (tX)) 
&= \int \limits_0^\infty \exp(tx) \cdot \text{Chi-Sq}(x | n) dx \\[8pt]
&= \frac{1}{2^{n/2} \Gamma(n/2)} \int \limits_0^\infty \exp(tx) \cdot x^{n/2-1} \exp(-x/2) dx \\[8pt]
&= \frac{1}{2^{n/2} \Gamma(n/2)} \int \limits_0^\infty x^{n/2-1} \exp((t -\tfrac{1}{2})x) dx. \\[8pt]
\end{aligned} \end{equation}$$
For the case where $t < \tfrac{1}{2}$, using the change-of-variable $r = (\tfrac{1}{2} - t)x$ we have:
$$\begin{equation} \begin{aligned}
M_X(t) 
&= \frac{1}{2^{n/2} \Gamma(n/2)} \int \limits_0^\infty x^{n/2-1} \exp((t -\tfrac{1}{2})x) dx. \\[8pt]
&= (\tfrac{1}{2} - t)^{-n/2} \cdot  \frac{1}{2^{n/2} \Gamma(n/2)} \int \limits_0^\infty r^{n/2-1} \exp(-r) dr. \\[8pt]
&= (1 - 2t)^{-n/2} \cdot \frac{1}{\Gamma(n/2)} \int \limits_0^\infty r^{n/2-1} \exp(-r) dr. \\[8pt]
&= (1 - 2t)^{-n/2}. \\[8pt]
\end{aligned} \end{equation}$$
For the case where $t \geqslant \tfrac{1}{2}$ we have:
$$\begin{equation} \begin{aligned}
M_X(t) 
&= \frac{1}{2^{n/2} \Gamma(n/2)} \int \limits_0^\infty x^{n/2-1} \exp((t -\tfrac{1}{2})x) dx. \\[8pt]
&\geqslant \frac{1}{2^{n/2} \Gamma(n/2)} \int \limits_0^\infty x^{n/2-1} dx. \\[8pt]
&= \frac{1}{2^{n/2} \Gamma(n/2)} \Bigg[ \frac{2}{n} x^{n/2} \Bigg]_{x=0}^{x \rightarrow \infty} \\[8pt]
&= \frac{1}{2^{n/2} \Gamma(n/2)} \cdot \infty = \infty. \\[8pt]
\end{aligned} \end{equation}$$
So, you can see that in this latter case, the integral defining the moment generating function diverges to infinity.  Because the value infinity is not in the set of real numbers, in such cases it is standard to say that the moment generating function "does not exist", meaning that there is no function $M_X: \mathbb{R} \rightarrow \mathbb{R}$ that satisfies the requirements over the argument values $t \geqslant \tfrac{1}{2}$.
