Let {$X_n$, n ${ge}$ 1} be a negatively associated sequence of identically distributed random variables with mean zeros and positive finite variances. Set $S_n$ = ${Sigma}^n_{i=1};X_i$. Suppose that 0 < ${sigma}^2=EX^2_1+2{Sigma}^{infty}_{i=2};Cov(X_1,;X_i)$ < ${infty}$. We prove that, if $EX^2_1(log^+{mid}X_1{mid})^{delta}$ < ${infty}$ for any 0< ${delta}{le}1$, then $lim_{{epsilon}downarrow0}{epsilon}^{2{delta}}sum_{{n=2}}^{infty}frac{(logn)^{delta-1}}{n^2}ES^2_nI({mid}S_n{mid}geq{epsilon}{sigma}sqrt{nlogn}=frac{E{mid}N{mid}^{2delta+2}}{delta}$, where N is the standard normal random variable. We also prove that if $S_n$ is replaced by $M_n=max_{1{le}k{le}n}{mid}S_k{mid}$ then the precise rate still holds. Some results in Fu and Zhang (2007) are improved to the complete moment case.