This maple worksheet produce Fourier sine expansions, plots, and approximations of functions on [0,1].

> with(plots):

input the number N of terms in the partial sums (use some number <20 to avoid long computations)

> N:=2; 

input the function  f(x)

> f:=x->x;

generate  and diaply the Fourier sine coefficients. of  f  up to N 

> b:=array(1..N);
> for n to N do b[n]:=2*int(sin(n*Pi*x)*f(x),x=0..1) od:

display the coefficients

> for n to N do b[n] od;


display the coefficients in decimal form

> for n to N do evalf(b[n]) od;

generate the partial sums of  f(x) 

> sf:=x->sum('evalf(b[j])*sin(j*Pi*x)','j'=1..N);

plot  f  and its partial Fourier sine sum up to N
and the periodic extension

> plot(f(x),x=0..1,title=`f(x)`,thickness=2);
> plot(sf(x),x=0..1, title=`Fourier approximation`,thickness=2);
> plot([f(x),sf(x)],x=0..1,title=`f and Fourier sine approxiamtion`,thickness=2);
> plot(sf(x),x=-5..5,title=`Periodic extension`,thickness=2);

compute f its partial sum and their diference at a given point x0

> x0:=.1;
> evalf(f(x0));
> evalf(sf(x0));
> evalf(f(x0)-sf(x0));