\[ \large \int_0^1 \int_0^1 \cdots \int_0^1 \left \{1 \div \prod_{n=1}^k x_n \right \} \, dx_1 \; dx_2 \; \cdots dx_k = 1 - \sum_{n=0}^{k-1} \dfrac{\gamma_n}{n!} \]

Prove that the equation above holds true where \( \gamma_n \) denotes the \(n^\text{th} \) Stieltjes constant, \(\displaystyle \gamma_n := \lim_{m\to\infty} \left [ \sum_{k=1}^m \dfrac{(\ln k)^n}k - \dfrac{(\ln m)^{n+1}}{n+1}\right ] \).

**Notation**: \( \{ \cdot \} \) denotes the fractional part function.

**Clarification**: There are \(k\) integrals.

This is a part of the set Formidable Series and Integrals

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