Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 19 / b / ii

The Newton root-finding iteration for this quadratic polynomial isAt $x_0=1$, it gives $x_1=1/(1-\epsilon )$. Its geometric series expansion is $1+\epsilon +{\epsilon }^2+{\epsilon }^3+O({\epsilon }^4)$. Subtracting the expansion of the polynomial root yields