n := (x,k)-> evalf(Re(sum(exp(I*'n^k'*x)/'n^4', n=2..20)))/(.08);f := (x,k)->cos(x)+.25*n(x,k);sf := (x,k,d) ->(sin(x+d)-sin(x))/d+.25*evalf(Re(sum((exp(I*'n^k'*(x+d))-exp(I*'n^k'*x))/(I*'n^(k+4)'), n=2..20))/(.08*d));plot([cos(x)],x=0..2*Pi);plot(n(x,12),x=0..2*Pi);plot(f(x,12),x=0..2*Pi);plot(sf(x,12,.0001),x=0..2*Pi);g := (x,e)-> piecewise(x<= -e, 0,x> -e and x<= 0, (x+e)/e^2, x>0 and x<= e, (e-x)/e^2,0);plot(g(x,.1),x=-1..1);fsg1 := (x,ee)-> (sf(x,12,ee)-sf(x,12,-ee))/ee;plot(fsg1(x,.01),x=0..2*Pi);fff := (x,k) -> -sin(x)+.25*evalf(Re(sum(exp(I*'n^k'*x)/(I*'n^(4+k)'), n=2..20)))/(.08);fff0 := (x,eee)->(evalf(Int(fff(x-y,12),y=0..eee))-evalf(Int(fff(x-y,12),y=-eee..0)))/eee^2;plot(fff0(x,.1),x=0..2*Pi);