  Let $\{X_n,n\geq 1\}$ be a sequence of i.i.d. random variables with partial
sums $\{S_n,n\geq 1\}$. Based on the classical Baum-Katz theorem, a paper by
Heyde in 1975 initiated the precise asymptotics for the sum $\sum_{n\geq
1}\mbox{P}(|S_n|\geq\epsilon n)$ as $\epsilon$ goes to zero. Later, Klesov
determined the convergence rate in Heyde's theorem. The aim of this paper is to
extend Klesov's result to the precise asymptotics for Davis law of large
numbers, a theorem in Gut and Sp\u{a}taru [2000a].
