# $\sum_{i=1}^{n}{\frac{x_i}{\sqrt{1-x_i}}} \geq \sqrt{\frac{n}{n-1}}$ for $x_i \in \mathbb{R}_{++}$ such that $\sum_{i=1}^{n}{x_i}=1$

Let $$x_1,\dots, x_n\in \mathbb{R}_{++}$$ such that $$\sum_{i=1}^{n}{x_i}=1$$. Prove that $$\sum_{i=1}^{n}{\frac{x_i}{\sqrt{1-x_i}}} \geq \sqrt{\frac{n}{n-1}}$$

I tried using AM-GM and Cauchy-Schwarz but didn't come to anything useful. Hint could be an help too.

The function $$f(x)=x/\sqrt{1-x}$$ is convex on $$(-1,1)$$, so by Jensen's inequality, one has \begin{align*} \sum\dfrac{1}{n}\dfrac{x_{i}}{\sqrt{1-x_{i}}}\geq \dfrac{\displaystyle\sum\dfrac{1}{n}x_{i}}{\sqrt{1-\displaystyle\sum\dfrac{1}{n}x_{i}}}. \end{align*}