Calculating $\sum_{n=1}^{\infty} r^n \cos n\theta$

Question

I'm struggling to calculate the sum of the following series: $$\sum_{n=1}^{\infty} r^n \cos n\theta$$ with $|r|<1$ and real, and $\theta \in [0,2 \pi] $ I'm not even sure if this is calculable. I've tried some tricks with the de Moivre formula and complex numbers, but I can't figure it out yet.

Answer 1

Note that your partial sum is the real part of the geometric sum $$\sum _{n=1}^{N}(re^{i\theta})^n$$
Compute this partial sum and use continuity of the real part function when taking the limit to deduce the answer (the answer will indeed be the real part of the limit of the complex geometric sum).