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?



Levi-Civita field in unusual basis
Let $p$ be any Laurent series in $\varepsilon$ of the form $\varepsilon^{-1}+\sum\_{n=0}^\infty a\_n\varepsilon^n$, like the one in the question. Then infinite Laurent series in $p$ itself never converge (not in the ring of Laurent series, or Levi-Civita field, or Hahn series, or any related such field), because positive powers of $p$ do not converge to zero.


On the other hand, if we consider $p^{-1}=\varepsilon+\sum\_{n=2}^\infty b\_n\varepsilon^n$, then this element topologically generates $\mathbb R[[\varepsilon]]$ (as a ring) and $\mathbb R((\varepsilon))$ (as a field): proving this comes down to the usual method of showing that you can iteratively find coefficients of a power series in $p^{-1}$ to make it give an arbitrary element of $\mathbb R[[\varepsilon]]$.


On the other hand, there is no way such a Laurent series in $p^{-1}$ can give you anything outside the ring of formal Laurent series. There is no way to produce fractional powers of $\varepsilon$ through this procedure.


