You are right István Sikari-Nágl, it is not as easy at all as it looks.
f[x_] = Piecewise[{{x + Pi, -Pi < x < 0}, {x - Pi, 0 < x < Pi}}]; g = FourierSeries[f[x], x, 2] // ExpToTrig (* -2 Sin[x] - Sin[2 x] *) Plot[{g, f[Mod[x, -2 \[Pi]]] + f[Mod[x, 2 \[Pi]]]}, {x, -3 \[Pi], 3 \[Pi]}, Exclusions -> None] 