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Convergence question

  1. Sep 20, 2010 #1
    Let 0<β<1

    So, β^n -> 0 as n -> infty
    Also, we can find γ>1 so that
    {γ^n}*{β^n} -> 0 as n -> infty
    e.g. γ = β^{-(1/2)}

    My question is how can i show that:
    n*β^n -> 0 as n -> infty
    and there exists γ>1 so that:
    {γ^n}*{n*β^n} -> 0 as n -> infty

    I appreciate any help
  2. jcsd
  3. Sep 20, 2010 #2
    Try showing that for any [tex]a>1[/tex] there exists an integer [tex]\bar n[/tex] such that

    [tex]n<a^n\qquad\textrm{for all}\qquad n>\bar n[/tex]

    To prove this you can use induction or differential calculus. After you have done this, you should be able to prove easily your claim.
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