How does one prove this equation?

Question

How does one prove the following equation , I am getting confused about this, I can't seem to find any proving technique, I tried plugging in the Stirling's formula for factorials but to no avail -

$\binom{2n}{n}$ = $\theta(4^n/n^(1/2))$

Answer 1

Per Stirling's formula,

$$k! \sim \sqrt{2\pi k} \left(\frac{k}{e}\right)^k,$$

we obtain

$$\binom{2n}{n} = \frac{(2n)!}{(n!)^2} \sim \frac{\sqrt{2\pi 2n}(2n)^{2n}e^{-2n}}{2\pi n (n^ne^{-n})^2} = \frac{2^{2n}}{\sqrt{\pi n}} = \frac{4^n}{\sqrt{\pi n}}.$$