Given the following cosine series
$ f(x) = \frac{1}{4} + \frac{4}{\pi^2} \sum_{n = 1}^{\infty} \frac{1}{n^2} (1-\cos(\frac{n \pi}{2}) ) \cos(\frac{n \pi x}{2}) $
Conjecture a possible closed form expression for the finction $f$ on (0,2) that is represented by this series by inspecting a few partial sum approximations.
So through using sympy, my closest plot was $f = 0.65 \cos(x)$ but it wasn't exact. Is there a more mathematical way to approach it to get a more exact form?
Using $$\sum\limits_{n=1}^\infty \frac{\sin(\pi n x)}{n}=\frac{\pi}{2}(1-x)$$ for $0<x<2$ one gets by integration $$\sum\limits_{n=1}^\infty \frac{\cos(\pi n x)}{n^2}= \frac{\pi^2}{6}-\frac{\pi^2}{4}(2x-x^2)$$. Use this to calculate $f(x)=1-x$, only elementary transformations are necessary.