Laurent series of $\frac{1}{(1-z)}$

Question

So I see this in my book:

So the first one I understand is the power series representation of a geometric series right?

But what is the second? Is the second just showing that this is an alternative representation of $\frac{1}{(1-z)}$ with negative powers and that's what makes it a laurent series? Since there are only negative powers in the second representation, that's what makes it a Laurent series right?

The two are different representations of the same series correct?

Answer 1

Basically correct. The two series have different regions of convergence: the first on $|z|\lt1$, the second $|z|\gt1$. They are both Laurent series. The first happens to be a Taylor series, as well. They are both around $z=0$.

I wouldn't say they are representations of the same series, but rather the same function on different regions.