Growth rate of a rank-one eigenvalue

Let $x_n = (1, 2, \dots, n)^\top \in \mathbb{R}^n$ and define the $n \times n$ matrix $A_n = x_n x_n^\top$. For each $n$, $A_n$ has exactly one nonzero eigenvalue $\lambda_n$. Find the integer $k$ for which $\dfrac{\lambda_n}{n^k}$ converges to a finite, nonzero limit as $n \to \infty$.