Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers:
Show that, for ##|r|<1,##
$$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$
My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above:
$$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$
The sum of this series is just:
$$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$
I'm having some trouble trying to figure out what to...