I just need to find a function representation for the following graph. I'm not sure why I can't figure it out, but nothing I've tried yields a correct series. :\
Specifically I need the range $\displaystyle -\pi \leq x \leq \pi$
I'm not terribly great with these types of functions... But I got:
$\displaystyle 10\left(\frac{x}{\pi} + 1\right) \ for \ -\pi \leq x < 0$
and
$\displaystyle \frac{10x}{\pi} \ for \ 0 < x \leq \pi$
I think I recognize the graph as a sawtooth wave... But I don't know how to formulate one...
What you have done is correct. All you must do now is integrate them to find the Fourier coefficients, and since your function is piecewise, you must split the integral into 2 integrals for each interval.
For example:
$\displaystyle A_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx = \frac{1}{\pi} \bigg( \int_{-\pi}^0 \big(10(\frac{x}{\pi} + 1\big)) \cos(nx) \, dx + \int_{0}^{\pi} 10\big(\frac{x}{\pi}\big) \cos(nx) \, dx \bigg) $