computation of the series $ \sum_{n=-\infty}^\infty \alpha ^n e^{in\theta}$

36 Views Asked by At

im having a hard time computing the following series:

$ \sum_{n=-\infty}^\infty f_n e^{in\theta}$

where $f_n=\alpha^n$, $f_n=f^*_{-n}$, $|\alpha|\leq 1$ is complex number and $f_0$ =1 . I can Splitt the series

$ \sum_{n=-\infty}^{-1} (\alpha^*)^n e^{in\theta} + \sum_{n=1}^\infty \alpha^n e^{in\theta} $

Im pretty sure that i need to use the geometrical series but i dont now what to do with $e^{in\theta} $