On the accuracy of approximation in the central limit theorem

Let
$$ \Delta_n=\sup_x|\mathbf P(\xi_1+\dots+\xi_n<x\sqrt n)-\Phi(x)|, $$
where $\xi_1,\xi_2,\dots$ are independent identically distributed random variables with the distribution function $F(x)$, $\mathbf E|\xi_1|^2=1$, $\mathbf E\xi_1=0$, and where $\Phi$ is the standard normal distribution function.
We investigate necessary and sufficient conditions on $F(x)$ for the following two series to converge:
$$ \sum h(\sqrt n)\frac{1}{n}\Delta_n<\infty,\quad\sum h(\sqrt n)n^{-3/2}\Delta_n<\infty, $$
where
$$ h(y)>0,\qquad h(y)\uparrow,\qquad h(y)/y\downarrow. $$
The case of Chebyshev–Gramer asymptotic expansions is also discussed.