Continuing from an earlier question of mine: Fourier-Series of a part-wise defined function?
I now got a fourier series which I believe is the correct one: $\frac{\pi(b-a)}{2} + \sum\limits_{n=1}^{\infty} \frac{(a-b)(1-(-1)^n)}{n^2\pi}\cos(nx) + \frac{(-1)^n(b-a)}{n}\sin(nx)$
Now my next task confused me a little - "What is the sum of this series?". By definition, this should just be $f(x)$ ($\frac{ax+bx}{2}$ if $x$ is a whole multiple of $\pi$) right? So I figured I am probably supposed to find a closed form for this - although given the definition of the function, I find it hard to imagine that it even exists. Am I wrong? If so, what is the closed form of this series?
(This wasn't the correct fourier series after all - the right one is
$\frac{\pi(b-a)}{2} + \sum\limits_{n=1}^{\infty} \frac{(a-b)(1-(-1)^n)}{n^2\pi}\cos(nx) + \frac{(-1)^{n+1}(a+b)}{n}\sin(nx)$