  For normalized sums $Z_n$ of i.i.d. random variables, we explore necessary
and sufficient conditions which guarantee the normal approximation with respect
to the R\'enyi divergence of infinite order. In terms of densities $p_n$ of
$Z_n$, this is a strengthened variant of the local limit theorem taking the
form $\sup_x (p_n(x) - \varphi(x))/\varphi(x) \rightarrow 0$ as $n \rightarrow
\infty$.
