The Asymptotic Behavior of a Family of Sequences via Tauberian Theorems

✍ Scribed by Patrick Martinez

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 158 KB
• Volume: 260
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.2001.7467

✦ Synopsis

We study the asymptotic behavior of a family of sequences defined by the following nonlinear induction relation c 0 = 1 and c n = k j=1 r j c n/m j + k j=k+1 r j c n+1 1/m j -1 for n ≥ 1, where the r j are real positive numbers and m j are integers greater than or equal to 2. Depending on the fact that k j=1 r j is greater or lower than 1, we prove that c n /n α or c n / ln n α goes to some finite limit for some explicit α. Our study is based on Tauberian theorems and extends a result of Erdös et al.