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Proof of convergence 
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#1
Nov1309, 09:34 PM

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P: 1,810

How can we prove that [tex]n^s(n1)^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
Nov1309, 09:53 PM

P: 1,060

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(n1)^s=s(n+\xi)^{s1}\to 0 [/tex] 


#3
Nov1309, 09:55 PM

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P: 1,810




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