Value of Apéry's Constant

Unsolved Problem
The infinite sum:
 * $\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^3}$

has not been determined.

By using the techniques of Basel Problem: Proof 1 it can be established that:


 * $\displaystyle \int_0^1 \int_0^1 \int_0^1 \frac {\mathrm d x \, \mathrm d y \, \mathrm d z} {1 - x y z} = \sum_{n \mathop = 1}^\infty \frac 1 {n^3}$