Basic math - normalizing the denominator

Question

I don't understand how to get from step 2 to step 3 here:

Is there some denominator normalization that I'm not getting? Can I use this method whenever I want to move a polynomial in a denominator to the numerator? Working with complex numbers here.

Answer 1

This is a calculation of the formal inverse of a series $$ 1 + ax + bx^2 + \cdots. $$ Multiplying out $$ (1 + ax + bx^2 + \cdots) (1 - ax + ?x^2 + \cdots) = 1 + 0x + \text{higher order terms} $$ shows that the coefficient of $x$ in the formal inverse is $-a$.

To prove convergence or find the rest of the coefficients you have to do the work in each particular case.