Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2009 / Paper 1 / 7 / b / Solution

Factor $1-z^3=(1-z)(1+z+z^2)$. The geometric series gives, for $|z|<1$,Thus the first three nonzero terms are $1-z+z^3$. If “first three terms” means the coefficients through degree two, that Taylor polynomial is $1-z+0z^2$. The distinction matters because the quadratic coefficient vanishes. The nearest singularities are the two nontrivial cube roots of unity, so the radius of convergence is exactly one.