Strange sum of Fourier series

Question

I have this function: $$g(x) = \begin{cases}1-\frac{2}{\pi^2}x^2 &, -\pi<x\le 0 \\\\ \cos(x) &, 0<x\le \pi\end{cases}$$ and its Fourier series: $$ \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx) ) $$

Now the test is asking me to evaluate the sum of this series: $$ \sum_{n=1}^{\infty} (-1)^n a_n $$

My prof. told me I have to use some properties and I need very little calculus. I thought I had to use Parseval but, honestly, I couldn't figure out how.

Can please somebody help me?

EDIT: I have the result if this can help: $=-\frac{7}{6}$

Answer 1

HINT:

Note that

$$\cos(n(\pm \pi))=(-1)^n$$

and that

$$g(\pm\pi)=-1$$

Now calculate $a_0$ and proceed.