Limited sum
Number Theory
Level
3
\[ \Large \lim_{x\to\infty} { \large \dfrac {\displaystyle \sum _{ l\le x }^{ }{ l \left\lfloor \frac { x }{ l } \right\rfloor } }{ { x }^{ 2 } } } \]
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 \( n\ge 1?\)
by
Nicolas Nagel