Other types of closed form for $\sum_{n=1}^\infty \frac{\cos(nx)}{n}$ in specific interval

Question

Is there other types of closed form for the following sum?

$$\sum_{n=1}^\infty \frac{\cos(nx)}{n}$$

needs to be valid over $ 0\le{x}\le{1}$

and need to be real numbers. The form below does not work for me, because its not valid in x=0.

$$-\ln\left(2\sin \frac {x}{2}\right) $$

Answer 1

for example, the summation below fig1

has a closed form valid in x=0. Does exist another closed form for the summation of the topic question, valid in x=0?

Answer 2

$\sum_{n\geq 1}\frac{\cos(nx)}{n}$ is converging by Dirichlet's test if $x\not\in 2\pi\mathbb{Z}$, since in such a case the partial sums of $\cos(nx)$ are bounded. With that assumption: $$ \sum_{n\geq 1}\frac{\cos(nx)}{n}=\text{Re}\sum_{n\geq 1}\frac{e^{nix}}{n} = -\text{Re}\log\left(1-e^{ix}\right) = -\log\|1-e^{ix}\| $$ hence: $$ \sum_{n\geq 1}\frac{\cos(nx)}{n}= -\frac{1}{2}\log\left((1-\cos x)^2+\sin^2 x\right) = -\frac{1}{2}\log\left(2-2\cos x\right)=\color{red}{-\log\left|2\sin\frac{x}{2}\right|} $$ that is not a continuous function over $\mathbb{R}$, as expected. Things change if you consider $$ \sum_{n\geq 1}\frac{\cos(nx)}{n^{\color{red}{2}}}, $$ since in such a case you get a periodic and continuous function given by parabolic arcs, because $$ \sum_{n\geq 1}\frac{\sin(nx)}{n}$$ is the Fourier series of the sawtooth wave.

The evaluation of the previous function at $x=\frac{\pi}{2}$ is one way to derive $\zeta(2)=\frac{\pi^2}{6}$, for instance.