State the sum of the series \(z+z^{2}+z^{3}+\ldots+z^{n}\), for \(z \neq 1\).

By letting \(z=\cos \theta+\mathrm{i} \sin \theta\), show that
\(\cos \theta+\cos 2 \theta+\cos 3 \theta+\ldots+\cos n \theta=\frac{\sin \frac{1}{2} n \theta}{\sin \frac{1}{2} \theta} \cos \frac{1}{2}(n+1) \theta\)
where \(\sin \frac{1}{2} \theta \neq 0\).