Let $X,X_1,X_2,;{cdots}$ be identically distributed and negatively associated random variables with mean zeros and positive, finite variances. We prove that, if $E{mid}X_1{mid}^r$ < ${infty}$, for 1 < p < 2 and r > $1+{frac{p}{2}}$, and $lim_{n{ ightarrow}{infty}}n^{-1}ES^2_n={sigma}^2$ < ${infty}$, then $lim_{{epsilon}{downarrow}0}{epsilon}^{{2(r-p}/(2-p)-1}{sum}^{infty}_{n=1}n^{{frac{r}{p}}-2-{frac{1}{p}}}E{{{mid}S_n{mid}}-{epsilon}n^{frac{1}{p}}}+={frac{p(2-p)}{(r-p)(2r-p-2)}}E{mid}Z{mid}^{frac{2(r-p)}{2-p}}$, where $S_n;=;X_1;+;X_2;+;{cdots};+;X_n$ and Z has a normal distribution with mean 0 and variance ${sigma}^2$.