Once floored, always floored

Calculus Level 4

Sn(x)=r=1nrx S_n(x) = \sum_{r=1}^{n} \lfloor rx \rfloor

For x x R \in \mathbb{R} , define Sn(x)S_n(x) as above.

Evaluate

limnSn(x)n2x=2015.20162017.\displaystyle \left\lfloor \left. \lim_{n \to \infty } \frac{S_n(x)}{n^2} \right|_{x = 2015.20162017} \right\rfloor.

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