\[ \large \int_0^1 \int_0^1 \cdots \int_0^1 \left\{ 1\div \sum_{n=1}^k x_n \right\} ^m \, dx_1 \; dx_2 \cdots dx_k \] Calculate in closed form, the integral above for \(k\geq 3\) and \(m\geq2 \).

**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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