Can all elements of the Levi-Civita field be represented as power series of a single element

$$p=\varepsilon^{-1}-\frac{\varepsilon }{24}+\frac{3 \varepsilon ^3}{640}-\frac{1525 \varepsilon ^5}{580608}+\dotsb$$

where the numerators of the terms are given in https://oeis.org/A118050 and the denominators are in https://oeis.org/A118051?

How would look $\varepsilon$ and $\varepsilon^{-1}$ in this basis?