Why do these laurent series approaches conflict? I was working on a problem of finding the Laurent series of $\frac{1}{z-3}$ that converges where $|z-4| > 1$
So I had one approach, let $u=z-4$ then:
$$\frac{1}{z-3} = \frac{1}{1+u} $$
$$ = \frac{1}{u} - \frac{1}{u^2} +  \frac{1}{u^3}...$$
$$ =  \frac{1}{z-4} -  \frac{1}{(z-4)^2} +  ...$$
But this apparently incorrect.
The correct answer is found by noting:
$$ \frac{1}{z-3} = \frac{1}{z-4 + 1} = \frac{1}{z-4} \frac{1}{1 - \frac{-1}{z-4}} = -\frac{1}{(z-4)^2} + ... $$
Where did I go wrong?

$$
\begin{align}
\frac1{z-4}\frac1{1+\frac1{z-4}}
&=\frac1{z-4}\left(1-\frac1{z-4}+\frac1{(z-4)^2}-\dots\right)\\
&=\frac1{z-4}-\frac1{(z-4)^2}+\frac1{(z-4)^3}-\dots
\end{align}
$$
The series starts with $\frac1{z-4}$, not $\frac1{(z-4)^2}$.
