Solution
= Solution
Write $1/f(x)=b_0+b_1x+b_2x^2+O(x^3)$ and multiply by $a_0+a_1x+a_2x^2+O(x^3)$. The product equals one, so its coefficients give $a_0b_0=1$, $a_0b_1+a_1b_0=0$ and $a_0b_2+a_1b_1+a_2b_0=0$. Since $a_0=f(0)\ne0$,
$$
\boxed{\frac1{f(x)}=\frac1{a_0}-\frac{a_1}{a_0^2}x+\frac{a_1^2-a_0a_2}{a_0^3}x^2+O(x^3).}
$$
This is the initial part of the general <reciprocal of a power series> recurrence.