h[x_,k_] = Sum[x^i,{i,0,k}];

Manipulate[Show[Plot[h[x,k],{x,-2,1},PlotRange->{{-1.5,1},{-0.5,10}},PlotStyle->Red],Plot[1/(1-x),{x,-2,1},PlotRange->{{-1.5,1},{-0.5,10}},PlotStyle->Black]
],
{k,0,20,1,Appearance->"Labeled"}
]

f[x_,k_] = Sum[(-1)^(i+1)*(x-1)^i/i,{i,1,k}];

Manipulate[Show[Plot[f[x,k],{x,0,5},PlotRange->{{0,2.5},{-2,2}},PlotStyle->Red],Plot[Log[x],{x,0,5},PlotRange->{{0,2.5},{-2,2}},PlotStyle->Black]
],
{k,0,20,1,Appearance->"Labeled"}
]