Plot[{Exp[x],Sum[x^n/n!, {n,0,3}]}, {x, -5, 5}]

Plot[{BesselJ[1,x],Sum[(-1)^n x^(2n+1)/(n!(n+1)!2^(2n+1)), {n, 0, 10}]}, {x, -10, 10}]

BesselPartialSum[n_,x_]:=Sum[(-1)^k x^(2k+1)/(k!(k+1)!2^(2k+1)), {k, 0, n}]

Manipulate[Plot[{BesselJ[1,x],BesselPartialSum[n,x]}, {x, -6, 6}, PlotRange->1],{n,0,10}]

TableForm[Table[{k,N[Abs[BesselJ[1,5]-BesselPartialSum[k,5]]]}, {k, 0, 10}]]

TableForm[Table[BesselPartialSum[k,x], {k, 0, 10}]]

PartialSum2[n_,x_]:=Sum[(k+1)(x-1)^k/2^(k+1),{k,0,n}]

Manipulate[Plot[{2/(3-x)^2,PartialSum2[n,x]},{x,-6,6},PlotRange->2],{n,0,10}]

TableForm[Table[PartialSum2[n,x],{n,0,10}]]

PartialSum2[4,0]

PartialSum3[n_,x_] := Sum[(-1)^k x^(2k+1)/(2k+1)!,{k, 0, n}]

TraditionalForm[TableForm[Table[PartialSum3[n,x],{n,0,10}]]]

Manipulate[Plot[{Sin[x],PartialSum3[n,x]},{x,-6,6},PlotRange->2],{n,0,10}]

Plot[Sum[(-1)^n x^(2n+1)/(2n+1)!,{n,0,10}],{x,-6,6}]

Plot[{BesselJ[1,x], Sum[(-1)^n x^(2n+1)/(n!(n+1)! 2^(2n+1)), {n, 0, 1}]},{x, -6, 6},PlotRange->1]