I'm trying to understand how the various different types of series with coefficients in $\mathbb{C}$ relate to each other, and have read that the field of Hahn series $\mathbb{C}(x^\mathbb{Z})$ is basically the same as the Laurent series $\mathbb{C}((x))$. Hahn series are described as a generalisation of Puiseux series, and Puiseux series have rational exponents, so if one looks at the Hahn series in $\mathbb{C}(x^\mathbb{Q})$ with rational exponents, how does this relate to the complex Puiseux series?
According to Wikipedia, the field of (complex) Puiseux series in the indeterminate $x$ is defined to be the series that have complex coefficients and rational exponents with bounded denominator and the field $\mathbb{C}(x^\mathbb{Q})$ of Hahn series consists of series of the form $$\sum_{q\in\mathbb{Q}}c_qx^q,$$ where $\lbrace q\in\mathbb{Q}:c_q\neq 0\rbrace$ is well-ordered with respect to the natural ordering on $\mathbb{Q}$ and $c_q\in\mathbb{C}$ i.e. complex coefficients and exponents are contained in a well-ordered subset of $\mathbb{Q}$. (Please correct me if I have misinterpreted these definitions!)
They cannot be exactly the same since $$x^{1/2}+x^{2/3}+x^{3/4}+x^{4/5}+\cdots$$ is in $\mathbb{C}(x^\mathbb{Q})$ but is not a Puiseux series because the denominators are unbounded. Am I missing something?
A Hahn series is not a Puiseux series because the denominators in the exponents is generally unbounded.