Does this limit hold? (with mean binomial-coefficient)

Question

Let $p\in[0,1]$ with $p\ne 1/2$. Does it hold that $$\lim_{n\to \infty} \binom{2n}{n}(1-p)^{n}p^{n}=0?$$

Answer 1

By ratio test

$$\frac{\binom{2n+2}{n+1}(1-p)^{n+1}p^{n+1}}{\binom{2n}{n}(1-p)^{n}p^{n}}=p(1-p)\frac{(2n+2)!n!n!}{(n+1)!(n+1)!(2n)!}=\\=p(1-p)\frac{(2n+2)(2n+1)}{(n+1)(n+1)}\to 4p(1-p)<1$$