I need to find the value of the following serie:
$ \sum_{n=0}^\infty \frac{1}{2^n}+\frac{(\pi i)^n}{n!}$
Our professor didn't show us how to do that with complex numbers.
$\frac{1}{2^n}$ gets smaller and smaller as n increases.
When I expand $\frac{(\pi i)^n}{n!}$, I get
$1+\pi i -\pi^2/2- i \pi^3/6 + \pi^4/24+ i\pi^5/120 - \pi^6/720 - ... $
As expected, the sign changes every two times because of the imaginary number, but I don't really see what to do next.
Thanks for your help.