A not so familiar summation

Calculus Level 5

\[ \large \lim\limits_{n \to \infty} \left( \frac{ \ln ( 1 + \frac{1}{n}) }{1} + \frac{ \ln (1 + \frac{2}{n})}{2} + \frac{ \ln(1 + \frac{3}{n})}{3} + \cdots + \frac{ \ln 2}{n} \right ) \]

If the limit above equals to \(A\), find \( \lfloor 10^4 A \rfloor \).

\(\)
Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.


Inspiration.

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