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I am searching for an explicit example of a sequence of real numbers $(u_n)$ that satisfies the following properties:

(1) for any $\alpha>0$, $n^{\alpha}\notin \mathcal{O}(u_n)$,

(2) There exists $\beta>0$ such that $u_n \notin \mathcal{O}(n^\beta)$.

(3) [Edited after comments] $(u_n)$ is non-decreasing

It might be easy, but I didn't find any of such $(u_n)$. For instance $\log(n)$ satisfies (1,3) but not (2). Could it be that (1,3) implies not (2) ?

Thanks

  • 0
    Just mix a sequence like $(\log n)$ with a sequence like $(n!)$ to get what you are looking for.2011-09-21
  • 0
    I don't see how that would give the desired behaviour, and what do you mean by mixing ?2011-09-21
  • 0
    Bounce back and forth from fast to slow: odd $n$ fast, even $n$ slow.2011-09-21
  • 0
    If you're careful you can even make it monotone.2011-09-21
  • 0
    @Yuval: How? I find that hard to believe.2011-09-21
  • 0
    Ok I see now, thanks ! But what if I want $(u_n)$ to be non-decreasing ? (Question edited)2011-09-21
  • 2
    $u_1 = 1$, then $u_2 = 2!$, then $u_{k+1} = (1 + 1/k^2) u_k$ until $u_k \le \log k$, then $u_{k+1} = (k+1)!$, etc.2011-09-21
  • 2
    For non-decreasing, slow many times, huge jump, slow many times, huge jump, and so on.2011-09-21

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