Chapter 7.3, Problem 13Q

Show that as $n\to \infty$, the pdf of a Student $t$ random variable with $n$ df converges to $f_Z\left(z\right)$. (Hint: To show that the constant term in the pdf for $T_n$ converges to $1/\sqrt{2\pi }$, use Stirling’s formula,

$n!=\sqrt{2\pi n}{n}^n{e}^{-n}$n

Also, recall that $\underset{n\to \infty }{\lim }{\left(1+\frac{a}{n}\right)}^n=e^a$.