## Solved – Limit of $t$-distribution as $n$ goes to infinity

I found in my intro to stats textbook that $t$-distribution approaches the standard normal as $n$ goes to infinity. The textbook gives the density for $t$-distribution as follows, $$f(t)=frac{Gammaleft(frac{n+1}{2}right)}{sqrt{npi}Gammaleft(frac{n}{2}right)}left(1+frac{t^2}{n}right)^{-frac{n+1}{2}}$$ I think it might be possible to show that this density converges (uniformly) to the density of normal as $n$ goes to infinity. Given $$lim_{nto infty}left(1+frac{t^2}{n}right)^{-frac{n+1}{2}}=e^{-frac{t^2}{2}}$$, … Read more