Show[
Plot[1/(1-x),{x,-4,1},PlotRange->{{-4,6},{-5,5}},PlotStyle->{Thick,Blue}],Plot[1/(1-x),{x,1,6},PlotRange->{{-4,6},{-5,5}}]
]

Manipulate[Show[Plot[Evaluate@Normal[Series[1/(1-z),{z,0,n}]],{z,-4,1.5},PlotRange->{{-4,2},{-5,5}},PlotStyle->{Red,Opacity[s]}],Plot[1/(1-z),{z,-4,1},PlotRange->{{-4,2},{-5,5}},PlotStyle->{Thick,Blue}]],{n,0,200,1},{s,0,1}]

Manipulate[Show[Plot[Evaluate@Normal[Series[Cos[z],{z,0,n}]],{z,-10,10},PlotRange->{{-10,10},{-2,2}},PlotStyle->{Red,Opacity[s]}],Plot[Cos[z],{z,-10,10},PlotRange->{{-10,10},{-1,1}},PlotStyle->{Thick,Blue}]],{n,0,30,1},{s,0,1}]