A Complicated Logarithmic Limit!

Calculus Level 5

$\large {x_n = \displaystyle \sum_{k=3}^n \sqrt{1 + \dfrac{1}{(k-1)^2} + \dfrac{1}{k^2}}}$

Let $x_n$ shown above be defined for $n$, where $n$ belongs to the set of natural numbers. Then find the value of ${ \displaystyle\lim_{n \to \infty}\dfrac{\ln(x_n)}{n}}$.

×