Taylors Series

This is a quick illustration for sqrt(x) about x = 1.

f[x_] = (Sqrt[x]); p[x_] = Normal[Series[f[x], {x, 1, 5}]]

Plot[{f[x], p[x]}, {x, 0, 3}];

Do[p[x_] = Normal[Series[f[x], {x, 1, n}]]; 
      Plot[{f[x], p[x]}, {x, 0, 3}, PlotRange -> {0, 2}], {n, 0, 20}]