SIMON-MARAIS 2020 A3

Determine the set of all real numbers $\alpha$ that can be expressed in the form

$$\alpha = \sum_{n=0}^{\infty} \frac{x_{n+1}}{x_n^3},$$

where $x_0, x_1, x_2, \ldots$ is an increasing sequence of real numbers with $x_0=1$.