De Rerum Natura

I have started to work thru Anton Deitmar’s “A First Course in Harmonic Analysis”. This is background reading for my perusal of the Compressed Sensing literature. On the first chapter he wants to prove the convergence in the $L^2$-norm of Fourier series for periodic functions. To do this he asserts (offhandedly to my untrained eyes) the equality
$$\sum _{k=1}^n\cos (2\pi kx)=\frac{\sin ((2n+1)\pi x)}{2\sin (\pi x)}–\frac{1}{2}.$$

It is embarrassing how long it has taken me to prove this on my own. In the end, Euler’s formula,
$$e^{i\theta }=\cos \theta +i\sin \theta ,$$
ended up being the right thing to use since it turns the proof into checking the equality of polynomials. My proof is inductive. First prove the $n=1$ case,
$$\cos (2\pi x)=\frac{\sin (3\pi x)}{2\sin (\pi x)}–\frac{1}{2},$$
by rewriting it as,
$$\sin (\pi x)\cos (2\pi x)=\frac{1}{2}(\sin (3\pi x)–\sin (\pi x))$$
$$\left(\frac{e^{i\theta }–e^{-i\theta }}{2i}\right)\left(\frac{e^{i2\theta }+e^{-i2\theta }}{2}\right)=\frac{1}{2}\left(\frac{e^{i3\theta }–e^{-i3\theta }}{2i}–\frac{e^{i\theta }–e^{-i\theta }}{2i}\right),$$
the demonstration of the equality becomes a polynomial check. Induction to the case $n+1$ by using the same trick with Euler’s formula leads to the general proof of the formula.