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limx→∞∑l≤xl⌊xl⌋x2 \Large \lim_{x\to\infty} { \large \dfrac {\displaystyle \sum _{ l\le x }^{ }{ l \left\lfloor \frac { x }{ l } \right\rfloor } }{ { x }^{ 2 } } } x→∞limx2l≤x∑l⌊lx⌋
Find the closed form of the above limit to 3 decimal places.
Notation: ⌊⋅⌋\lfloor \cdot \rfloor ⌊⋅⌋ denotes the floor function.
Bonus: Can you find a general formula for limx→∞∑l≤xln⌊xl⌋xn+1\displaystyle \lim _{ x\to \infty}{ \frac {\sum _{ l\le x }^{ }{ {l}^{n} \left\lfloor \frac { x }{ l } \right\rfloor } }{ { x }^{ n+1 } } }x→∞limxn+1∑l≤xln⌊lx⌋ for n≥1? n\ge 1?n≥1?
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