<s>[INST] Is this asymptotic equation correct? Is this equation correct?
$$ \frac {1 + \Theta(\frac 1 {2n})} {(1 + \Theta(1/n))^2} = 1 + O(1 / n) $$
I need this equation to prove that 
$$ \binom {2n} n = \frac {2 ^ {2n}} {\sqrt {\pi n}} (1 + O(1 / n)) $$
which could be calculated from Stirling's Approximation:
$$ n! = \sqrt {2\pi n} \left(\frac n e\right)^n \left(1 + \Theta({\frac 1 n})\right).$$
 [/INST] Suppose $Bx\le\Theta(x)\le Cx$ for each $\Theta$.
$$
\begin{align}
\frac{1+\Theta(\frac{1}{2n})}{(1+\Theta(\frac{1}{n}))^2}
&\le\frac{1+\frac{C}{2n}}{(1+\frac{B}{n})^2}\\
&=(1+\frac{C}{2n})(1-2\frac{B}{n}+O(\frac{1}{n^2}))\\
&=(1+(\frac{C}{2}-2B)\frac{1}{n}+O(\frac{1}{n^2}))\\
&=1+O(\frac{1}{n})
\end{align}
$$
$$
\begin{align}
\frac{1+\Theta(\frac{1}{2n})}{(1+\Theta(\frac{1}{n}))^2}
&\ge\frac{1+\frac{B}{2n}}{(1+\frac{C}{n})^2}\\
&=(1+\frac{B}{2n})(1-2\frac{C}{n}+O(\frac{1}{n^2}))\\
&=(1+(\frac{B}{2}-2C)\frac{1}{n}+O(\frac{1}{n^2}))\\
&=1+O(\frac{1}{n})
\end{align}
$$
</s>