I am trying to work on problem 5.18 on Casella Berger and have a doubt. I will write the main informations below and then will point out where I am having trouble.
Let $X$ be a random variable with a Student's $t$ distribution with $p$ degrees of freedom. Show that:
\begin{align} \lim_{p\rightarrow\infty} f(x|p) \rightarrow \frac{1}{\sqrt{2\pi}}e^{-x^2/2} \end{align}
And the book hints on using Stirling's Formula:
\begin{align} n!\approx\sqrt{2\pi}n^{n+1/2}e^{-n} \end{align}
Now, the distribution of a Student's $t$ random variable is: \begin{align} f(x|p)=\frac{\Gamma(\frac{p+1}{2})}{\Gamma{\frac{p}{2}}}\frac{1}{\sqrt{p\pi}}\frac{1}{(1+\frac{x^2}{p})^{(p+1)/2}} \end{align}
To use Stirling's Formula in this context it becomes pretty obvious to me that I have to use what I understand to be the definition of $\Gamma(n)$ if $n$ is an integer: \begin{align} \Gamma(n) = (n-1)! \end{align}
The problem is that $\frac{p+1}{2}$ and $\frac{p}{2}$ cannot both be integers. So Stirling's Formula will only work for one of them. Now, if I assume that the formula will work for both of them I can work out the limit to the correct answer. But is this a valid assumption? Am I missing some Gamma Function property here?
Thanks in advance