A Complicated Logarithmic Limit!

Calculus Level 4

xn=k=3n1+1(k1)2+1k2\large {x_n = \displaystyle \sum_{k=3}^n \sqrt{1 + \dfrac{1}{(k-1)^2} + \dfrac{1}{k^2}}}

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

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