Let {X, X n , n ≥ 1} be a sequence of i.i.d. random variables with EX = 0, EX 2 = σ 2. Set S n = n k=1 X k and let N be the standard normal random variable. Let g(x) be a positive and twice differentiable function on [n 0 , ∞) such that g(x) ∞, g (x) 0 as x → ∞. In this short note, under some suitable conditions on both X and g(x), we establish the following convergence rates in precise asymptotics lim ε 0 ∞ n=n 0 g (n)P |S n | σ √ n ≥ εg s (n)-ε-1/s E|N | 1/s = γη, where γ = lim n→∞ (n k=n 0 g (k)-g(n)), η = ∞ n=n 0 g (n)P{S n = 0}. It can describe the relations among the boundary function, weighting function, convergence rate and the limit value in studies of complete convergence. The result extends and generalizes the corresponding results of Gut and Steinebach (Ann. Univ. Sci.