Manipulate[Plot[{(2/Pi)*(∑_(n=1)^N)Sin[n*Pi*x]((-1)^(n+1))/n,-1,1},{x,-1.1,1.1}],{N,1,100,1}]

Plot[{-1,1,Evaluate@Table[(2/Pi)*(∑_(n=1)^N)Sin[n*Pi*x]((-1)^(n+1))/n,{N,1,41,16}]},{x,-1.1,1.1}]

Plot[{1,Evaluate@Table[(2/Pi)*(∑_(n=1)^N)Sin[n*Pi*x]((-1)^(n+1))/n,{N,1,81,8}]},{x,0.9,1.1},PlotRange->{0.5,1.2}]