a monotone sequence

Question

Let $x_n = \left( 1+ \frac{1}{n} \right)^{n + \alpha}$ be a sequence where $n \in \mathbb{N^*}$. Show that for $\alpha$$\lt$$\frac 12$ ,$\exists$ k$\in \mathbb{N^*}$ such that $\lor n\geq k$, we have $x_n\lt x_{n+1}$.
I showed that for $\alpha= \frac 12$, $x_n$ is a decreasing sequence starting with $n=1$ but i don't know what to do next.