Let X 1 , X 2 , ⋯ {X_1},{X_2}, \cdots be i.i.d. random variables and let S n = X 1 + ⋯ + X n {S_n} = {X_1} + \cdots + {X_n} . The relationship between the tth moment of X 1 {X_1} and the convergence of the series ∑ n = 1 ∞ z n n t − 1 P ( S n > 0 ) \sum \nolimits _{n = 1}^\infty {{z^n}{n^{t - 1}}P({S_n} > 0)} is investigated in this paper. The convergence of the series above when | z | = 1 |z| = 1 but z ≠ 1 z \ne 1 is related to the oscillation of the sequence { P ( S n > 0 ) } \{ P({S_n} > 0)\} and to the oscillation of the sequence { S n } \{ {S_n}\} about zero.