Code:

f[t_] := Which[-6 <= t <= -5, -t - 5, -3 <= t <= -2, t + 3,
-2 <= t <= -1, -t - 1, 1 <= t <= 2, t - 1, 2 <= t <= 3,
-t + 3, 5 <= t <= 6, t - 5, -5 <= t <= -3, 0, -1 <= t <= 1,
0, 3 <= t <= 5, 0];
fplot = Plot[f[x], {x, -6, 6}];
a0 = NIntegrate[f[t], {t, -6, 6}];
a = Table[(1/6)*NIntegrate[f[t]*Cos[((Pi*n)/6)*t], {t, -6, 6}],
{n, 1, 25}];
b = Table[(1/6)*NIntegrate[f[t]*Sin[((Pi*n)/6)*t], {t, -6, 6}],
{n, 1, 25}];
fs[x_] := N[a0/12 + Sum[a[[n]]*Cos[Pi*(n/6)*x] +
b[[n]]*Sin[Pi*(n/6)*x], {n, 1, 25}]];
fsplot = Plot[fs[x], {x, -6, 6}, PlotStyle -> Red];
Show[{fplot, fsplot}]