Why do we choose $\frac{1}{1-z} = 1 + z + z^2 + \cdots$ instead of $\frac{1}{1-z} = z + z^2 +\cdots$?

Question

In Laurent series, we have the following

$$\frac{1}{1-z} = 1 + z + z^2 + z^3 + \dots + z^n, \qquad 0 < \mid z\mid < 1, \quad n \rightarrow \infty$$

However, $\frac{1}{1-z} = z + z^2 + z^3 ... +z^n (0 < \mid z\mid < 1, n \rightarrow \infty)$ also works.

Since geometric series can be written

$$z + z^2 + z^3 ... + z^n = \frac{1 - z^n}{1-z} (0 < \mid z\mid < 1, n \rightarrow \infty)$$ $$z + z^2 + z^3 ... + z^n = \frac{1}{1-z} (0 < \mid z\mid < 1, n \rightarrow \infty,\mid z^n \mid \rightarrow 0) $$

My question is: why do we not use $$\frac{1}{1-z} = z + z^2 + z^3... + z^n (0 < \mid z\mid < 1, n \rightarrow \infty) ?$$

Answer 1

This equation you wrote is obviously a MISTAKE: $$z+z^2+z^3..+z^n=\frac{1-z^n}{1-z}.$$ Why?

Put $z=0$. Then:

$$0+0+0...+0=\frac{1-0^n}{1-0}\implies 0=1.$$