Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 15 / b / Solution

Since $a_0=f(0)\ne 0$, write $1/f(x)=b_0+b_{1}x+b_2{x}^2+O(x^3)$ and multiply the two power series. Equating coefficients with the identity $f(x)/f(x)=1$ givesSolving successively yieldsThese are the constant, linear and quadratic terms; particular coefficients may vanish. For a convergent power series defining $f$ near zero, continuity and $a_0\ne 0$ ensure its reciprocal is also defined in a neighborhood of zero.