assume (L>0);
f := x->piecewise(
 0 <= x and x<= L/2, x,
 L/2 <= x and x<=L,  -x+L);
a0 := 2/L*int(f(x),x=0..L);
an := 2/L*int(f(x)*cos(n*Pi*x/L),x=0..L);
simplify(an) assuming n::even;
simplify(an) assuming n::odd;
parsum := (x,N) -> 1/4*L + sum((-1+(-1)^n)*4*L/((2*n)^2*Pi^2)*cos(2*n*Pi*x/L),n=1..N);
L:=10; 
plot({f(x),parsum(x,1)},x=0..L);
plot({f(x),parsum(x,2)},x=0..L);
plot({f(x),parsum(x,3)},x=0..L);
plot({f(x),parsum(x,4)},x=0..L);
plot({f(x),parsum(x,10)},x=0..L);