# Find value of serie with complex number

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.

• $e^z=\sum \frac {z^n}{n!}$ even for complex $z$. – lulu Feb 21 '18 at 19:22