It is well-known that the complete convergence theorem for i.i.d. random variables has been an active topic since the famous work done by Hsu and Robbins . Chow  obtained a moment version of Hsu and Robbins series. However, the series tends to infinity whenever ε goes to zero, so it is of interest to investigate the asymptotic behavior of the series as ε goes to zero. This note gives some limit theorems of the series generated by moments for NA random variables.