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Limit of sequence differences

  1. Oct 24, 2012 #1
    We know that for [itex]k\in\mathbb{N}[/itex] we have, if:

    [itex]\displaystyle\lim_{n\to\infty}\left(a_{n}-a_{n-k}\right)=k\cdot a[/itex]

    then:

    [itex]\displaystyle\lim_{n\to\infty}\frac{a_n}{n}=a[/itex]

    When the reverse impliaction is also true? What do we have to assume to achieve if and only if theorem? I'm especially interested in a case when [itex]a=0[/itex]. I state (now without a proof, just intuition) that if

    [itex]\displaystyle\lim_{n\to\infty}\frac{a_n}{n}=a[/itex]

    then [itex]\displaystyle\limsup_{n\to\infty}\left(a_{n}-a_{n-k}\right)[/itex] and [itex]\displaystyle\liminf_{n\to\infty}\left(a_{n}-a_{n-k}\right)[/itex] are bounded and if these limits are the same, they are equal [itex]0[/itex].
     
    Last edited: Oct 24, 2012
  2. jcsd
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