Connecting formal Laurent series to complex analysis?

From what I have read and this MSE question, the formal Laurent series $\in \Bbb{K}(\!(x)\!)$ can be thought of as (semi-)infinite sequences written in the form:
$$\sum^\infty_{n=d|d\ne-\infty}a_nx^n\quad \text{where}\quad a_n \in\Bbb{K}$$

with the following designation of notation:
\[\begin{aligned}(a,0,0,0...)\quad& \text{denoted by}& a=ax^0\\
(0,a,0,0...)\quad& \text{denoted by}& ax=ax^1\\
(0,0,a,0...)\quad &\text{denoted by}& ax^2 \end{aligned}\]
_{I will denote this type of Laurent series FLS (formal Laurent series).}

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In complex analysis however the Laurent series of a function $f(z)$ is given by:
$$f(z)=\sum_{k=-\infty}^\infty a_k(z-a)^k$$

where:
$$a_k=\frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^{k+1}}dz$$
This is defined on an annulus about $a$ in which $f(z)$ is analytic.

_{I will denote this type of Laurent series CLS (complex Laurent series).}

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As we can see there are some difference between these the FLS and CLS:

1. The FLS lives in the space of (semi-)infinite sequences whilst the CLS lives in the space of functions $f:\Bbb{C}\rightarrow \Bbb{C}$.



2. The FLS is not permitted to have an infinite lower bound whilst the CLS does.


3. For CLS we can come we can come up with a property of convergence whilst for the FLS we can't.

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My Question: How are the FLS and CLS linked, in such away that reconciles there differences? i.e. what is the connection between them.

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_{If you could provide references, if you used them, that would be great.}