# Limited sum

$\Large \lim_{x\to\infty} { \large \dfrac {\displaystyle \sum _{ l\le x }^{ }{ l \left\lfloor \frac { x }{ l } \right\rfloor } }{ { x }^{ 2 } } }$

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 $\displaystyle \lim _{ x\to \infty}{ \frac {\sum _{ l\le x }^{ }{ {l}^{n} \left\lfloor \frac { x }{ l } \right\rfloor } }{ { x }^{ n+1 } } }$ for $n\ge 1?$

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