Something like this:
f[x_] = x;
fcap[0] =
With[{n = 0}, Integrate[f[t]/E^(I*n*t), {t, -Pi, Pi}]/(2*Pi)];
fcap[n_ /; n != 0] =
Simplify[Integrate[f[t]/E^(I*n*t), {t, -Pi, Pi}]/(2*Pi),
Element[n, Integers]];
partialSum[n_, x_] = 0 + Sum[2*Re[fcap[k]*E^(I*k*x)], {k, 1, n}];
Manipulate[Plot[partialSum[n, x], {x, -Pi, Pi}], {n, 0, 15, 1}]
