f[x_] := Log[1 + x]
p[x_,n_] := (∑_(i=1)^n)(((((-1))^(i-1))(x^i))/i)

Manipulate[Plot[{f[x],p[x,n]},{x,-.8,1.2},PlotRange-> {-1.2,1.2}, PlotStyle->Thick],{n,1,40,1}]

e[n_] := (1/((2n + 1)!))

Plot[e[n],{n,0,15}]

N[e[3]]