Solution
= 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 yields
$$
\boxed{S=\cos\frac{15\pi}{7}=\cos\frac\pi7.}
$$
The upper summation limit matters: the first fourteen terms complete one full cycle, and the fifteenth supplies one additional term.