Dirichlet kernel in Fourier series partial sum

P[u_,M_]:=((Sin[(M+1/2)u])/(Sin[u/2]))

Manipulate[Plot[P[u,M],{u,-Pi,Pi},PlotRange->{{-Pi,Pi},{-30,120}},PlotPoints->10000],{M,1,60,1}]

Manipulate[Plot[P[u,M],{u,-Pi,Pi},PlotRange->{{Pi/2-.1,Pi/2+.1},{-10,10}},PlotPoints->10000],{M,1,500,1}]