***************************************************

p0[x_] = 1; 
p1[x_] = 1 + x; 
p2[x_] = 1 + x + x^2/2; 
p3[x_] = 1 + x + x^2/2 + x^3/6; 
p4[x_] = 1 + x + x^2/2 + x^3/6 + x^4/24; 
p5[x_] = 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120; 
Plot[{E^x, p0[x], p1[x], p2[x]}, {x, -4, 4}, 
   AspectRatio -> 1.5, PlotRange -> {-4, 10}]; 
Plot[{E^x, p2[x], p3[x], p4[x], p5[x]}, {x, -4, 4}, 
   AspectRatio -> 1.5, PlotRange -> {-4, 10}];

***************************************************

p0[x_] = 1; 
p1[x_] = 1 + (x - 1)/2; 
p2[x_] = 1 + (x - 1)/2 - 1/8*(x - 1)^2; 
p3[x_] = 1 + (x - 1)/2 - 1/8*(x - 1)^2 + 1/16*(x - 1)^3; 
p4[x_] = 1 + (x - 1)/2 - 1/8*(x - 1)^2 + 1/16*(x - 1)^3 - 
    5/128*(x - 1)^4; 
p5[x_] = 1 + (x - 1)/2 - 1/8*(x - 1)^2 + 1/16*(x - 1)^3 - 
    5/128*(x - 1)^4 + 1/256*(x - 1)^5; 
Plot[{Sqrt[x], p1[x], p2[x]}, {x, 0, 4}, PlotRange -> {0, 3}, 
   PlotStyle -> 
    {GrayLevel[0.], GrayLevel[0.6], GrayLevel[0.6]}]; 
  Plot[{Sqrt[x], p3[x], p4[x], p5[x]}, {x, 0, 4}, 
   PlotRange -> {0, 3}, PlotStyle -> 
    {GrayLevel[0.], GrayLevel[0.6], GrayLevel[0.6], 
     GrayLevel[0.6]}];

Plot[{Sqrt[x], p2[x]}, {x, 0, 4}, PlotRange -> {0, 3}];

Plot[{Sqrt[x], p3[x]}, {x, 0, 4}, PlotRange -> {0, 3}];

Plot[{Sqrt[x], p4[x]}, {x, 0, 4}, PlotRange -> {0, 3}];

Plot[{Sqrt[x], p5[x]}, {x, 0, 4}, PlotRange -> {0, 3}];

***************************************************

Clear[f, p]
f[x_] = Log[1 + x]; 
p[x_, n_] = Sum[((-1)^(k + 1)*x^k)/k, {k, 1, n}];

Plot[{Log[1 + x], p[x, 1]}, {x, -0.99, 3}, 
   PlotRange -> {-5, 6}];

Plot[{Log[1 + x], p[x, 2]}, {x, -0.99, 3}, 
   PlotRange -> {-5, 6}];

Plot[{Log[1 + x], p[x, 5]}, {x, -0.99, 3}, 
   PlotRange -> {-5, 6}];

Plot[{Log[1 + x], p[x, 10]}, {x, -0.99, 3}, 
   PlotRange -> {-5, 6}];

***************************************************

Clear[f, p]
f[x_] = E^x; 
p[x_, n_] := Sum[x^k/k!, {k, 0, n}]

Plot[{E^x, p[x, 1]}, {x, -5, 5}, PlotRange -> {-5, 20}];

Plot[{E^x, p[x, 2]}, {x, -5, 5}, PlotRange -> {-5, 20}];

Plot[{E^x, p[x, 3]}, {x, -5, 5}, PlotRange -> {-5, 20}];

Plot[{E^x, p[x, 5]}, {x, -5, 5}, PlotRange -> {-5, 20}];

Plot[{E^x, p[x, 10]}, {x, -5, 5}, PlotRange -> {-5, 20}];

Plot[{E^x, p[x, 15]}, {x, -5, 5}, PlotRange -> {-5, 20}];

Plot[{E^x, p[x, 15]}, {x, -10, 10}, PlotRange -> {-10, 40}];

***************************************************

Clear[f, p, x, n, a, i]; f[x_] = Sqrt[1 + x]; 
  Do[a[i] = D[f[x], {x, i}]/i! /. x -> 0, {i, 0, 50}]
Clear[poly, n, x]; poly[n_] := Sum[a[i]*x^i, {i, 0, n}]

Do[Plot[{f[x], poly[n]}, {x, -1, 3}, PlotRange -> {0, 3}, 
   PlotStyle -> 
    {GrayLevel[0.], GrayLevel[0.6], GrayLevel[0.6]}, 
   AxesOrigin -> {0, 0}], {n, 1, 8}]

***************************************************

Clear[f, p, x, n, a, i]; f[x_] = Sin[x]; 
  Do[a[i] = D[f[x], {x, i}]/i! /. x -> 0, {i, 0, 50}]
Clear[poly, f, n, x]; poly[n_] := Sum[a[i]*x^i, {i, 0, n}]

Do[Plot[{Sin[x], poly[4*n]}, {x, -20, 20}, 
   PlotRange -> {-2, 2}, PlotStyle -> 
    {GrayLevel[0.], GrayLevel[0.6], GrayLevel[0.6]}, 
   AxesOrigin -> {0, 0}], {n, 1, 12}]