I have a distribution function of random value $F_N (x)=Φ(x)+\frac{3}{50\sqrt{2π}} e^{-x^2/2} (x^3-3x) \frac {6N^3+21N^2+31N+31} {N(2N+5)^2 (N-1)}+O(\frac{1}{N^2} )$. The function depends on $N>2$ and it was derived as asymptotic expansion in the Central Limit Theorem. Due to the asymptotic expansion there is an error like $O(\frac{1}{N^2})$
I need to calculate pdf and I am not sure how to do it. What will happens with the error of the approximation in pdf?