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 \({ \displaystyle\lim_{n \to \infty}\dfrac{\ln(x_n)}{n}}\).

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