I'm starting with complex numbers and a friend has sent me an exercise to try to do, but I feel completely lost while thinking about it. Could you give me a hand with this? Thanks!
Be $\rho \in R$ with $|\rho| < 1$ and $ \phi \in R$. Calculate $\sum_{n=0}^{\infty} \rho^n\cos(n\phi)$ and $\sum_{n=0}^{\infty} \rho^n\sin(n\phi)$.
\begin{align}\sum_{n=0}^\infty\rho^n\cos(n\phi)+i\sum_{n=0}^\infty\rho^n\sin(n\phi)&=\sum_{n=0}^\infty\rho^n\left(\cos(n\phi)+i\sin(n\phi)\right)\\&=\sum_{n=0}^\infty\left(\rho e^{i\phi}\right)^n.\end{align}Can you take it from here?