<s>[INST] 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?
 [/INST] ${1\over 1+x} = 1 - x + x^2 - x^3 + \cdots$ when $|x| < 1$.
This is derived from ${1 \over 1-x} = 1 + x + x^2 + \cdots$ with $|x| < 1$.
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