If the middle term of ${\left(1+x\right)}^{2n}$  is the greatest term, then $x$  lies between

$T_{n+1}$  has greatest binomial coefficient ${}^{2n}{\text{C}}_n$

$\therefore \frac{\left(2n+1\right)\left|x\right|}{\left|x\right|+1}=x+F\Rightarrow x<\frac{\left(2n+1\right)\left|n\right|}{\left|n\right|+1}<n+1$

$\Rightarrow \frac{n}{2n+1}<\frac{\left|x\right|}{\left|x\right|+1}<\frac{n+1}{2n+1}$

$\Rightarrow \frac{n}{2n+1}<1-\frac{1}{\left|x\right|+1}<\frac{n+1}{2n+1}$

$\frac{-\left(n+1\right)}{2n+1}<-\frac{1}{\left|x\right|+1}<\frac{-n}{2n+1}$

$\frac{2n+1}{n+1}<\left|x\right|+1<\frac{2n+1}{n}$

$\frac{n}{n+1}<\left|x\right|<\frac{n+1}{n}$