What is $\cos x-\cos2x+\cos3x-\cos4x...\pm\cos(Nx)$?

Question

I want to arrive at a closed expression for $f_N(x)=\frac{2}{π}(\cos x-\cos2x+\cos3x-\cos4x...\pm \cos Nx)$ ($+\cos(Nx)$ if $N$ is odd, $-\cos(Nx)$ if $N$ is even)

Using the fact that $\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$ and $\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$, and by expressing our function as a G.P. with common ratio $-e^{-ix}$, I was able to express $f_N(x)$ (when $N$ is odd) as:

$$(\frac{1}{π}•\frac{e^{Nix}-e^{-Nix}}{1+e^{-ix}})+\frac{1}{π}$$

However I got stuck when I tried to simplify the above equation in terms of trigonometric terms. I was only able to simplify upto:

$$\frac{-1}{π}•\frac{\sin(Nx)•2i}{1+e^{-ix}}+\frac{1}{π}$$

I was a bit worried I had done something wrong while expressing the function as a G.P. but I checked my work and it seems fine.

I guess that the answer lies in some simple manipulation of terms but unfortunately, I was not able to find the solution to this problem.

Appreciate any help :)

Answer 1

Expanding on @Bernard's answer, for even $N$ the sum is$$\begin{align}\Re\left[e^{ix}\frac{1-e^{iNx}}{1+e^{ix}}\right]&=\Re\left[e^{ix}\frac{-2ie^{iNx/2}\sin\frac{Nx}{2}}{2\cos\frac{x}{2}e^{ix/2}}\right]\\&=\Im\left[e^{i(N+1)x/2}\frac{\sin\frac{Nx}{2}}{\cos\frac{x}{2}}\right]\\&=\frac{\sin\frac{Nx}{2}\cos\frac{(N+1)x}{2}}{\cos\frac{x}{2}},\end{align}$$while for odd $N$ it's$$\Re\left[e^{ix}\frac{1+e^{iNx}}{1+e^{ix}}\right]=\Re\left[e^{i(N+1)x/2}\frac{\cos\frac{Nx}{2}}{\cos\frac{x}{2}}\right]=\frac{\cos\frac{Nx}{2}\cos\frac{(N+1)x}{2}}{\cos\frac{x}{2}}.$$You can unify these e.g. as$$\frac{\sin\left(\frac{Nx}{2}+(1+(-1)^{N+1})\frac{\pi}{4}\right)\cos\frac{(N+1)x}{2}}{\cos\frac{x}{2}}.$$

Answer 2

Hint:

Here is a fast method:

This alternating sum is the real part of

$$\sum_{k=1}^N (-1)^{k-1}\mathrm e^{ikx}=\mathrm e^{ix}\sum_{\ell=0}^{N-1} (-1)^{\ell}\mathrm e^{i\ell x}=\mathrm e^{ix}\frac{1-(-1)^N\mathrm e^{iNx}}{1+\mathrm e^{ix}}.$$ Now in the numerator of the fraction, factor out $\:\mathrm e^{\tfrac{iN}2}$, and in the denominator, $\:\mathrm e^{\tfrac{ix}2}$. Can you proceed?