Onlyperiodic functions have Fourier series. (For non-periodic functions, you use Fourier transforms).

You want a Fourier series for the single periodic function

f(x)= x for

f(x)= 1 for

and "extended by periodicity": f(x+ 2)= f(x).

Use the usual formulas for the Fourier coefficients. You will need to break the integral into two parts, from 0 to 1 and from 1 to 2.