Proof of convergence

  1. disregardthat

    disregardthat 1,841
    Science Advisor

    How can we prove that [tex]n^s-(n-1)^s[/tex] converge to zero as [tex]n \to \infty[/tex] where s as a real number satisfies [tex]0<s<1[/tex]?

    I am specifically looking for a more or less elementary proof for this for real s. I think we can use the infinite binomial expansion, but I am looking for something that does not require more than elementary calculus.
     
  2. jcsd
  3. I thought about the graph of that function and why the limit is "obvious" from the graph. Translating the graph picture into mathematics, I think an easy way is the mean value theorem.
    With it you find
    [tex]
    n^s-(n-1)^s=s(n+\xi)^{s-1}\to 0
    [/tex]
     
  4. disregardthat

    disregardthat 1,841
    Science Advisor

    Excellent, nice and easy proof!
     
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