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# Integrals!

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

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

Note by Hummus A
1 year, 9 months ago

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