Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 1 / 13 / c / Solution

Put $d=a-b$. The rational factor has Taylor expansion $1+dx-bd{x}^2+b^{2}d{x}^3+O(x^4)$. Multiplication by the logarithmic power series makes the coefficients of $x^2,x^3$ equal to $d-1/2$ and $1/3-d/2-bd$. Setting both to zero gives $d=1/2$ and $b=1/6$, henceThe coefficient of $x^4$ is $-1/4+d/3+bd/2+b^{2}d=-1/36$. Thus the resulting expansion begins $x-x^4/36+O(x^5)$.