Central binomial coefficients

Stirling. $(2n)!\sim\sqrt{4\pi n}(2n/e)^{2n}$ and $(n!)^2\sim2\pi n(n/e)^{2n}$.

Ratio. $\binom{2n}{n}=\frac{(2n)!}{(n!)^2}\sim\frac{\sqrt{4\pi n}\cdot 4^n n^{2n}/e^{2n}}{2\pi n\cdot n^{2n}/e^{2n}}=\frac{2\sqrt{\pi n}}{2\pi n}\cdot 4^n=\frac{4^n}{\sqrt{\pi n}}$.

Conclusion. Central binomials grow like $4^n/\sqrt{\pi n}$.

Remark. This is used to prove $\sum\frac{1}{n\binom{2n}{n}}$ converges.