The Asymptotic Behavior of a Family of Sequences via Tauberian Theorems

We study the asymptotic behavior of a family of sequences defined by the following nonlinear induction relation c0 = 1 and cn = k j=1 rjc n/mj + k j=k+1 rjc n+1 1/mj −1 for n ≥ 1, where the rj are real positive numbers and mj are integers greater than or equal to 2.Depending on the fact that k j=1 rj is greater or lower than 1, we prove that cn/nα or cn/ ln n α goes to some finite limit for some explicit α.Our study is based on Tauberian theorems and extends a result of Erd¨os et al.