# LogicalExpand to find coefficients in power series

I am attempting to use the LogicalExpand command to find an equation for each coefficient in a power series. The documentation gives the following example of this usage:

In:= LogicalExpand[Series[g[x]^2,{x,0,2}]==2+x]

Out= -2+g^2==0&&-1+2 g g'==0&&g'^2+g g''==0


The goal is to get a system of equations like the one above corresponding to the coefficients of each power of $$x$$ in the power series.

My series is more complicated than the one in the documentation, and it should be identically equal to $$0$$, so that each coefficient should be identically zero. When I apply the LogicalExpand function, however, Mathematica simply returns the original input rather than a system of equations. The expression to which I am trying to apply logical expand is: $$\frac{1}{6} x \left(f'(0) h''(0)+h'(0) \left(f'(0) \left(2 h'(0)+3\right)+3\right)+h''(0)+6\right)+\frac{1}{12} x^2 \left(f''(0) \left(h''(0)+h'(0) \left(2 h'(0)+3\right)\right)-2 f'(0)^2 \left(h''(0)+h'(0) \left(4 h'(0)+3\right)\right)-4 f'(0) \left(h''(0)+h'(0) \left(h'(0)+3\right)+3\right)\right)=0$$

In Mathematica:

In:=LogicalExpand[
1/6 x (6 + h' (3 + f' (3 + 2 h')) + h'' + f' h'') +
1/12 x^2 (-4 f' (3 + h' (3 + h') + h'') + f'' (h' (3 + 2 h') + h'') -
2 f'^2 (h' (3 + 4 h') + h'')) == 0]

Out:=
1/6 x (6 + h' (3 + f' (3 + 2 h')) + h'' + f' h'') +
1/12 x^2 (-4 f' (3 + h' (3 + h') + h'') + f'' (h' (3 + 2 h') + h'') -
2 f'^2 (h' (3 + 4 h') + h'')) == 0


I would appreciate any thoughts on why this is failing. Thank you!

Try with ordersymbol O[x] :
LogicalExpand[1/6 x (6 + h' (3 + f' (3 + 2 h')) + h'' +f' h'') +1/12 x^2 (-4 f' (3 + h' (3 + h') + h'') +f'' (h' (3 + 2 h') + h'') -2 f'^2 (h' (3 + 4 h') + h'')) == O[x]^3]