Abstract: Based on the previous documentary,when the real random variable sequence satisfies the definite condition,the complete convergence is discussed.And this condition is the order of their finite moment which is less than one,but the problem of the convergent rate can't be finished absolutely.In this paper,the convergent rates of the random variabe sequence are point out.At last,when the order of their finite moment is less than one,the result obtained by Xue Liugen is generalized to Banach space and the results are also shown in this paper.