Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2003 / Paper 1 / 8 / b / Solution

Write $1/f(x)=b_0+b_{1}x+b_2{x}^2+O(x^3)$ and multiply by $a_0+a_{1}x+a_2{x}^2+O(x^3)$. The product equals one, so its coefficients give $a_0{b}_0=1$, $a_0{b}_1+a_1{b}_0=0$ and $a_0{b}_2+a_1{b}_1+a_2{b}_0=0$. Since $a_0=f(0)\ne 0$,This is the initial part of the general reciprocal of a power series recurrence.