I just watched the Vimeo video "What is Pi and what it is not" where Doron Zeilberger talks about $\pi$. At around 16:00, he writes this on the blackboard: $$\pi = \lim_{n} \frac{4^n}{{2n \choose n}n}$$ But that doesn't work. As $n$ increases, this expression seems to tend towards $0$.
Does anyone have a clue as to what he might have meant to write? I'm guessing the expression needs only a little adjustment.
Stirling's Approximation for the factorial is $$n! = \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n (1 + R_1(n)) ,$$ where $R_1(n) \in O\left(\frac1n\right)$. Substituting gives that $${2 n \choose n} = \frac{(2n)!}{n!^2} = \frac{4^n}{\sqrt{\pi n}} \left(1 + R_2(n)\right),$$ where again $R_2(n) \in O\left(\frac1n\right)$. Rearranging gives the limit $$\boxed{\lim_{n \to \infty} \frac{4^n}{\sqrt{n} {{2 n} \choose n}} = \sqrt{\pi}} .$$