The Power of Complex Numbers

I've reached a part of my learning which starts to touch upon exactly why I signed up for this module. Had to share...

How would you normally ever go about finding the sum of this infinite series of natural numbers?

\sum^{\infty}_{n=0}\frac{1}{2^{n}}\sin nx


Well! Complex numbers to the rescue!

\sum^{\infty}_{n=0}\frac{1}{2^{n}}\sin nx



=\ldots (simplification steps)

=\frac{2\sin x}{5-4\cos x}

How cool is that!