Proof of convergence


by disregardthat
Tags: convergence, proof
disregardthat
disregardthat is offline
#1
Nov13-09, 09:34 PM
Sci Advisor
P: 1,686
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.
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Gerenuk
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#2
Nov13-09, 09:53 PM
P: 1,057
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]
disregardthat
disregardthat is offline
#3
Nov13-09, 09:55 PM
Sci Advisor
P: 1,686
Quote Quote by Gerenuk View Post
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]
Excellent, nice and easy proof!


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