f1[x_]:=Sin[1*x];
f2[x_]:=Sin[2*x];
f3[x_]:=Sin[3*x];
tf={f1[x],f2[x],f3[x]};
Plot[tf,{x,0,1}]

F[k_,x_]:=Sin[k*x];
tF=Table[F[k,x],{k,25}];
Plot[tF,{x,0,1}]

sumF[x_]:=F[1,x]+F[2,x]+F[3,x];
Plot[{F[1,x],F[2,x],F[3,x],sumF[x]},{x,0,1}]

tF=Table[F[k,x],{k,25}];
sumF[x_]:=Sum[F[k,x],{k,1,500}];
Plot[tF,{x,0,1}];
Plot[sumF[x],{x,0,1}]