Past exam of the mathematical methods I course of the University of Oxford / 2014 / 6 / Solution

Let $\omega =e^{i\pi /7}$, a primitive fourteenth root of unity. The geometric series gives ${\sum }_{k=1}^{14}{\omega }^k=0$. Taking the real part and adding the remaining fifteenth term therefore yieldsThe upper summation limit matters: the first fourteen terms complete one full cycle, and the fifteenth supplies one additional term.