# series(y,x,x0,n) is series expansion of y with x at x=x0 for n degree;
#series(y_,x_,x0_,n_):= sum(d(y,x,x0,k)*(x-x0)^k/k!,k,0,n,1);
#series(y_,x_,n_):=series(y,x,0,n);
#series(y_,x_):=series(y,x,0,8);
#series(y_):=series(y,x,0,8);


series(integrate(y_,x_)):=integrate(series(y),x);

series( erf(x_)):= 2 x/sqrt(pi) - 2 x^3/(3 sqrt(pi)) + x^5/(5 sqrt(pi));

series( asin(x_)):= x + x^3/6 + 3 x^5/40;
series( acos(x_)):= pi/2-x - x^3/6 - 3 x^5/40;
series( tan(x_)):= x + x^3/3 + 2 x^5/15;
series( cot(x_)):= 1/x - x/3 - x^3/45 - 2 x^5/945;
series( sec(x_)):= 1+x^2/2 + 5x^4/24;
series( csc(x_)):= 1/x + x/6 + 7x^3/360 + 31 x^5/15120;
series( sin(x_)):= x - x^3/6 +  x^5/120;
series( cos(x_)):= 1-1/2*x^2 + x^4/24 ;

series(a_*b_):=expand(series(a)*series(b));
series(a_+b_):=series(a)+series(b);