Consider the following function and series:

$$ f(z)=-\frac{1}{z} \frac{1-\frac{z^2}{2!}+O(z^4)}{1-\frac{z^2}{3!}+O(z^4)} \tag{1}.$$

I've seen on many posts here and in textbooks that the following can be done:

$$ f(z)=-\frac{1}{z} \left ( 1-\frac{z^2}{2!}+O(z^4) \right) \left ( 1+\frac{z^2}{3!}+O(z^4) \right)\tag{2}$$

Why can the denominator be brought up with changed signs like that?


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