Function for a cosine series

Question

I have a cosine series. When I plot I notice it can converge. I will be glad to know the simple function $f(x)$ for this Fourier series. Thanks. $$f(x)=\sum_1^\infty{(-1)^n \over n}\cos (nx)$$ I find a similar Fourier series, but it is a sine series. $f(x)=x=2\sum_1^\infty{(-1)^{(n+1)} \over n}\sin(nx)$

Answer 1

\begin{align*} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}\mathrm{cos}(nx)&=\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}(e^{inx}+e^{-inx})/2\\ &=\frac{1}{2}\sum_{n=1}^{\infty}\frac{(-e^{ix})^{n}}{n}+\frac{1}{2}\sum_{n=1}^{\infty}\frac{(-e^{-ix})^{n}}{n}\\ &=-\frac{1}{2}\ln(1+e^{ix})-\frac{1}{2}\ln(1+e^{-ix})\\ &=-\frac{1}{2}\ln(2+2\cos(x)). \end{align*}