c[n_]=FourierSinCoefficient[1,x,n]

DiscretePlot[c[n],{n,1,22},PlotRange->All,AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[β, j])}]

d[n_]=-c[n]/n^2;

DiscretePlot[d[n],{n,1,6},PlotRange->All,AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,Ticks->{{1,2,4,6},{-0.5,-1}},AxesLabel->{j,(Subscript[ξ, j])}]

s1[x_]=FourierSinSeries[1,x,1];

Plot[{s1[x], 1},{x,0,Pi},PlotRange->{0,1.45},PlotStyle->Thick]

Plot[{s1[x],1,c[3]*Sin[3x]},{x,0,Pi},PlotRange->{-.45,1.45},PlotStyle->Thick]

s3[x_]=FourierSinSeries[1,x,3];

Plot[{s3[x],1},{x,0,Pi},PlotRange->{0,1.45},PlotStyle->Thick]

Plot[{s3[x],1,c[5]*Sin[5x]},{x,0,Pi},PlotRange->{-.45,1.45},PlotStyle->Thick]

s5[x_]=FourierSinSeries[1,x,5];

Plot[{s5[x],1},{x,0,Pi},PlotRange->{0,1.45},PlotStyle->Thick]

Plot[{s5[x],1,c[7]*Sin[7x]},{x,0,Pi},PlotRange->{-.45,1.45},PlotStyle->Thick]

s7[x_]=FourierSinSeries[1,x,7];

Plot[{s7[x],1},{x,0,Pi},PlotRange->{0,1.45},PlotStyle->Thick]

Plot[{s7[x],1,c[9]*Sin[9x]},{x,0,Pi},PlotRange->{-.45,1.45},PlotStyle->Thick]

s9[x_]=FourierSinSeries[1,x,9];

Plot[{s9[x],1},{x,0,Pi},PlotRange->{0,1.45},PlotStyle->Thick]

s21[x_]=FourierSinSeries[1,x,21];

Plot[{s21[x],1},{x,0,Pi},PlotRange->{0,1.45},PlotStyle->Thick]

Plot[{s1[x],1,s5[x],s21[x]},{x,0,Pi},PlotRange->{0,1.45},PlotStyle->Thick]

u1[x_]=FourierSinSeries[x*(x-Pi)/2,x,1];

Plot[{u1[x],x*(x-Pi)/2},{x,0,Pi},PlotRange->{-1.35,0},PlotStyle->Thick]

Plot[{u1[x],x*(x-Pi)/2,d[3]*Sin[3x]},{x,0,Pi},PlotRange->{-1.35,.1},PlotStyle->Thick]

u3[x_]=FourierSinSeries[x*(x-Pi)/2,x,3];

Plot[{u3[x],x*(x-Pi)/2},{x,0,Pi},PlotRange->{-1.35,0},PlotStyle->Thick]

Plot[{u3[x],x*(x-Pi)/2,d[5]*Sin[5x]},{x,0,Pi},PlotRange->{-1.35,.1},PlotStyle->Thick]

u5[x_]=FourierSinSeries[x*(x-Pi)/2,x,5];

Plot[{u5[x],x*(x-Pi)/2},{x,0,Pi},PlotRange->{-1.35,.1},PlotStyle->Thick]

Plot[{Sin[x],Sin[2x],Sin[3x],Sin[4x]},{x,0,Pi},PlotStyle->Thick]