Answer 1

It looks odd, could you give the example you want, bcz in matlab finding fournier series of a coef is like:
function[ak] = cal_fs(x,w0,N)
ak = zeros(1,2*N+1);
T=2*pi/w0;
syms t;
ak(1,1+k+N) = 1/T* int(x*exp(-1i*k*w0*t),t,0,T); % ak : fournier coef
end