1

I have seen somewhere that if $0\le \alpha<1$, then zero is a cluster point of the sequence $n^\alpha \sin n, n=1,2,\cdots$.

My question is what if $\alpha=1$? Or $\alpha>1$?

TCL
  • 14,262
  • If $\alpha \geq 1$ look at the limit of the sequence. – Girish May 13 '13 at 14:45
  • @Girish: Are you claiming that the limit exists? – Jonas Meyer May 13 '13 at 14:49
  • 2
    See http://math.stackexchange.com/q/221018/. Not exactly a duplicate, but this question is contained in that one. – Jonas Meyer May 13 '13 at 14:51
  • @ Jonas i should have said
    If α≥1 look at the limit of the sequence if it exists.     – Girish May 13 '13 at 15:01
  • 2
    For $\alpha=1$, this is just the question of whether the continued fraction coefficients for $\pi$ are unbounded. I don't think we know the answer to that question. – Thomas Andrews May 13 '13 at 15:01
  • 1
    I'm fairly ignorant about such questions, but looking at the answers in the question linked to by Jonas Meyer, it looks as if this is simply too hard to answer with certainty from the current state of knowledge. Just in case anybody has something helpful to answer here, I can add some related questions: is it known whether that sequence has any cluster points at all? And in case it should have cluster points, is it in any way more or less likely that $0$ is one of them (this is horribly inprecise!). – Marc van Leeuwen May 13 '13 at 15:13

0 Answers0