Bounding a Sequence

For each integer $n \ge 2$, define $$x_n = n\left(\sqrt[n]{2} - 1\right) - \left(\frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{n+n}\right).$$ Find the largest constant $\alpha$ such that $x_n > \alpha$ for all $n \ge 2$.