Uniform Convergence

Some examples: sequences of functions and uniform convergence.

gn[n_,x_]:=(1-x^(2n))/(1+x^(2n))

gn[5,x]

Plot[{gn[1,x],gn[6,x],gn[25,x]},{x,-2,2}]

Plot[Piecewise[{{-1,x<-1},{1,-1<x<1},{-1,x>1},{0,x==1}}],{x,-2,2}]

Plot[{Piecewise[{{-1,x<-1},{1,-1<x<1},{-1,x>1},{0,x==1}}],gn[1,x],gn[6,x],gn[20,x]},{x,-.9,.9},PlotRange->{-1,1}]

Power Series and uniform convergence

ff[x_]:=Exp[x];polyn[x_]=Series[ff[x], {x, 0, 3}]//Normal;Plot[{polyn[x],ff[x]},{x,-4,4},PlotRange->{-20,60}]