f[x_]:=Exp[-100*(x-1.5)^2];

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

DiscretePlot[c[n],{n,1,22},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[β, j])}]

DiscretePlot[Exp[-n^2*0.01]*c[n],{n,1,22},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

DiscretePlot[Exp[-n^2*0.02]*c[n],{n,1,22},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

DiscretePlot[Exp[-n^2*0.03]*c[n],{n,1,22},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

DiscretePlot[Exp[-n^2*0.05]*c[n],{n,1,22},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

DiscretePlot[Exp[-n^2*0.1]*c[n],{n,1,22},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

DiscretePlot[Exp[-n^2*0.2]*c[n],{n,1,22},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

DiscretePlot[Exp[-n^2*0.3]*c[n],{n,1,22},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

DiscretePlot[Exp[-n^2*0.5]*c[n],{n,1,22},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

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

Plot[Sum[c[n]*Sin[n*x],{n,1,25}],{x,0,Pi},PlotRange->{0,1},PlotStyle->Thick]

Plot[{Sum[Exp[-n^2*0.03]*c[n]*Sin[n*x],{n,1,10}],Exp[-0.03]*c[1]*Sin[x],Exp[-2^2*0.03]*c[2]*Sin[2*x],Exp[-3^2*0.03]*c[3]*Sin[3*x],Exp[-4^2*0.03]*c[4]*Sin[4*x],Exp[-5^2*0.03]*c[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.1,0.3},PlotStyle->Thick]

t=0.05;

Plot[{Sum[Exp[-n^2*t]*c[n]*Sin[n*x],{n,1,10}],Exp[-t]*c[1]*Sin[x],Exp[-2^2*t]*c[2]*Sin[2*x],Exp[-3^2*t]*c[3]*Sin[3*x],Exp[-4^2*t]*c[4]*Sin[4*x],Exp[-5^2*t]*c[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.1,0.3},PlotStyle->Thick]

t=0.1;

Plot[{Sum[Exp[-n^2*t]*c[n]*Sin[n*x],{n,1,10}],Exp[-t]*c[1]*Sin[x],Exp[-2^2*t]*c[2]*Sin[2*x],Exp[-3^2*t]*c[3]*Sin[3*x],Exp[-4^2*t]*c[4]*Sin[4*x],Exp[-5^2*t]*c[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.1,0.3},PlotStyle->Thick]

t=0.2;

Plot[{Sum[Exp[-n^2*t]*c[n]*Sin[n*x],{n,1,10}],Exp[-t]*c[1]*Sin[x],Exp[-2^2*t]*c[2]*Sin[2*x],Exp[-3^2*t]*c[3]*Sin[3*x],Exp[-4^2*t]*c[4]*Sin[4*x],Exp[-5^2*t]*c[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.1,0.3},PlotStyle->Thick]

t=0.5;

Plot[{Sum[Exp[-n^2*t]*c[n]*Sin[n*x],{n,1,10}],Exp[-t]*c[1]*Sin[x],Exp[-2^2*t]*c[2]*Sin[2*x],Exp[-3^2*t]*c[3]*Sin[3*x],Exp[-4^2*t]*c[4]*Sin[4*x],Exp[-5^2*t]*c[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.1,0.3},PlotStyle->Thick]

t=0.3;

Plot[{Sum[Exp[-n^2*t]*c[n]*Sin[n*x],{n,1,10}],Exp[-t]*c[1]*Sin[x],Exp[-2^2*t]*c[2]*Sin[2*x],Exp[-3^2*t]*c[3]*Sin[3*x],Exp[-4^2*t]*c[4]*Sin[4*x],Exp[-5^2*t]*c[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.1,0.3},PlotStyle->Thick]

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

e[n_]=Exp[-n^2*0.05]*c[n];

DiscretePlot[e[n],{n,1,10},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[β, j])}]

t=0.2;

DiscretePlot[Cos[n*t]*e[n],{n,1,10},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

t=0.4;

DiscretePlot[Cos[n*t]*e[n],{n,1,10},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

t=0.6;

DiscretePlot[Cos[n*t]*e[n],{n,1,10},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

t=0.8;

DiscretePlot[Cos[n*t]*e[n],{n,1,10},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

t=1;

DiscretePlot[Cos[n*t]*e[n],{n,1,10},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

t=1.2;

DiscretePlot[Cos[n*t]*e[n],{n,1,10},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

t=1.4;

DiscretePlot[Cos[n*t]*e[n],{n,1,10},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

t=1.6;

DiscretePlot[Cos[n*t]*e[n],{n,1,10},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

t=1.8;

DiscretePlot[Cos[n*t]*e[n],{n,1,10},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

t=2;

DiscretePlot[Cos[n*t]*e[n],{n,1,10},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

t=2.2;

DiscretePlot[Cos[n*t]*e[n],{n,1,10},PlotRange->{-.12,.12},AxesOrigin->{1,0},PlotMarkers->Automatic,Joined->True,AxesLabel->{j,(Subscript[ξ, j])}]

t=0;

Plot[{Sum[Cos[n*t]*e[n]*Sin[n*x],{n,1,10}],Cos[t]*e[1]*Sin[x],Cos[2*t]*e[2]*Sin[2*x],Cos[3*t]*e[3]*Sin[3*x],Cos[4*t]*e[4]*Sin[4*x],Cos[5*t]*e[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.15,0.25},PlotStyle->Thick]

t=0.2;

Plot[{Sum[Cos[n*t]*e[n]*Sin[n*x],{n,1,10}],Cos[t]*e[1]*Sin[x],Cos[2*t]*e[2]*Sin[2*x],Cos[3*t]*e[3]*Sin[3*x],Cos[4*t]*e[4]*Sin[4*x],Cos[5*t]*e[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.15,0.25},PlotStyle->Thick]

t=0.4;

Plot[{Sum[Cos[n*t]*e[n]*Sin[n*x],{n,1,10}],Cos[t]*e[1]*Sin[x],Cos[2*t]*e[2]*Sin[2*x],Cos[3*t]*e[3]*Sin[3*x],Cos[4*t]*e[4]*Sin[4*x],Cos[5*t]*e[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.15,0.25},PlotStyle->Thick]

t=0.6;

Plot[{Sum[Cos[n*t]*e[n]*Sin[n*x],{n,1,10}],Cos[t]*e[1]*Sin[x],Cos[2*t]*e[2]*Sin[2*x],Cos[3*t]*e[3]*Sin[3*x],Cos[4*t]*e[4]*Sin[4*x],Cos[5*t]*e[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.15,0.25},PlotStyle->Thick]

t=0.8;

Plot[{Sum[Cos[n*t]*e[n]*Sin[n*x],{n,1,10}],Cos[t]*e[1]*Sin[x],Cos[2*t]*e[2]*Sin[2*x],Cos[3*t]*e[3]*Sin[3*x],Cos[4*t]*e[4]*Sin[4*x],Cos[5*t]*e[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.15,0.25},PlotStyle->Thick]

t=1;

Plot[{Sum[Cos[n*t]*e[n]*Sin[n*x],{n,1,10}],Cos[t]*e[1]*Sin[x],Cos[2*t]*e[2]*Sin[2*x],Cos[3*t]*e[3]*Sin[3*x],Cos[4*t]*e[4]*Sin[4*x],Cos[5*t]*e[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.15,0.25},PlotStyle->Thick]

t=1.2;

Plot[{Sum[Cos[n*t]*e[n]*Sin[n*x],{n,1,10}],Cos[t]*e[1]*Sin[x],Cos[2*t]*e[2]*Sin[2*x],Cos[3*t]*e[3]*Sin[3*x],Cos[4*t]*e[4]*Sin[4*x],Cos[5*t]*e[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.15,0.25},PlotStyle->Thick]

t=1.4;

Plot[{Sum[Cos[n*t]*e[n]*Sin[n*x],{n,1,10}],Cos[t]*e[1]*Sin[x],Cos[2*t]*e[2]*Sin[2*x],Cos[3*t]*e[3]*Sin[3*x],Cos[4*t]*e[4]*Sin[4*x],Cos[5*t]*e[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.15,0.25},PlotStyle->Thick]

t=1.6;

Plot[{Sum[Cos[n*t]*e[n]*Sin[n*x],{n,1,10}],Cos[t]*e[1]*Sin[x],Cos[2*t]*e[2]*Sin[2*x],Cos[3*t]*e[3]*Sin[3*x],Cos[4*t]*e[4]*Sin[4*x],Cos[5*t]*e[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.15,0.25},PlotStyle->Thick]

t=1.8;

Plot[{Sum[Cos[n*t]*e[n]*Sin[n*x],{n,1,10}],Cos[t]*e[1]*Sin[x],Cos[2*t]*e[2]*Sin[2*x],Cos[3*t]*e[3]*Sin[3*x],Cos[4*t]*e[4]*Sin[4*x],Cos[5*t]*e[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.15,0.25},PlotStyle->Thick]

t=2;

Plot[{Sum[Cos[n*t]*e[n]*Sin[n*x],{n,1,10}],Cos[t]*e[1]*Sin[x],Cos[2*t]*e[2]*Sin[2*x],Cos[3*t]*e[3]*Sin[3*x],Cos[4*t]*e[4]*Sin[4*x],Cos[5*t]*e[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.15,0.25},PlotStyle->Thick]

t=2.2;

Plot[{Sum[Cos[n*t]*e[n]*Sin[n*x],{n,1,10}],Cos[t]*e[1]*Sin[x],Cos[2*t]*e[2]*Sin[2*x],Cos[3*t]*e[3]*Sin[3*x],Cos[4*t]*e[4]*Sin[4*x],Cos[5*t]*e[5]*Sin[5*x]},{x,0,Pi},PlotRange->{-0.15,0.25},PlotStyle->Thick]