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!

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

$=\text{Im}\left(\sum^{\infty}_{n=0}\frac{1}{2^{n}}e^{inx}\right)$

$=\text{Im}\left(\frac{1}{1-\frac{1}{2}e^{ix}}\right)$

$=\ldots$ (simplification steps)

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

How cool is that!

Adrian