1. ## Geometric Series

Use complex numbers and geometric series to sum:

$1 + cos2\theta + cos4\theta +...+ cos(2n-2)\theta$.

I don't really have a clue what to do... any help would be appreciated... thanks. The answer is $\frac{\cos{(n-1)\theta} \sin\: {n\theta}}{\sin{\theta}}$.

2. Consider $\Re (1 + e^{2i\theta}\ + e^{4i\theta}\ + ... + e^{2(n - 1)i\theta})\$ .

3. Hello

The key point is to use de Moivre's formula: $[\cos(x)+i\cdot{\sin(x)}]^k=\cos(kx)+i\cdot{\sin(kx)}$
And mix it with the geometrical sum

http://en.wikipedia.org/wiki/De_Moivre's_formula

PS. It can also be done using telescoping sums

4. Just be careful what Simba posted does not work for $\theta = \pi n$. (Because that is the special case of geometric series which fails).