We consider the Lindeberg-Feller model for independent random variables and focus our attention on the behaviour of the probability densities q_n of sums S_n, n\geq 1 . We obtain a theorem on the convergence of q_n to the standard normal density \varphi which resembles the well known limit theorem for distribution functions--provided that the q_n are positive definite. A special case is the following: if q_n(0)\rightarrow\varphi(0) as n\rightarrow\infty then the Lindeberg condition guarantees that the convergence of q_n to \varphi continues to the real line.