"Transseries" are a kind of generalized power series that allow things like fractional exponents and exponentials (with another transseries as the exponent). I know very little about them but I have looked through e.g. Edgar's Transseries for beginners. The transseries considered there are "expansions around infinity", so that the powers of the variable $x$ must decrease to the right; thus for instance the usual power series $\sum_{n=0}^\infty \frac{x^n}{n!}$ for $e^x$ is not a transseries (at $\infty$), but we do I think have a transseries
$$ \sum_{n=0}^\infty \frac{1}{n! x^n}$$
that is a sort of power series at $\infty$ for $e^{1/x}$.
On the other hand, unless I am confused, $e^{1/x}$ is also itself a transseries. So it seems that in the world of transseries we have two different ways to represent the same function $f(x)=e^{1/x}$, as "$e^{1/x}$" and as the above series "$ \sum_{n=0}^\infty \frac{1}{n! x^n}$".
My question is, assuming this is correct, how should I think about this redundancy? Is it an unavoidable disadvantage of the extra expressivity of transseries over power series? Or is there some positive reason why we want to distinguish "$e^{1/x}$" and "$ \sum_{n=0}^\infty \frac{1}{n! x^n}$"?
