Proving Convergence: Real s, 0<s<1, n \to \infty

In summary, the conversation discusses how to prove that the expression n^s-(n-1)^s converges to zero as n approaches infinity, where s is a real number between 0 and 1. The speaker is specifically looking for an elementary proof that does not require advanced calculus. They suggest using the mean value theorem to show that the expression ultimately converges to zero. Another participant agrees that this is a simple and effective proof.
  • #1
disregardthat
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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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  • #2
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]
 
  • #3
Gerenuk said:
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!
 

1. How do you prove the convergence of a real series with a limit of 0?

To prove the convergence of a real series with a limit of 0, we use the limit comparison test, which compares the given series with a known series that converges. If the two series have a similar behavior, then the given series also converges.

2. What does it mean for a series to converge?

A series converges when the sum of its terms approaches a finite value as the number of terms increases. In other words, the series has a finite limit as the number of terms approaches infinity.

3. Why is it important to prove the convergence of a series?

Proving the convergence of a series is important because it ensures that the sum of its terms is well-defined and has a finite value. This allows us to use the series in calculations and make conclusions based on its behavior.

4. What is the significance of the real number s in proving convergence?

The real number s represents the power of the terms in the series. For a series to converge, s must be greater than 0 and less than 1. If s is outside of this range, the series will either diverge or have an indeterminate limit.

5. How does the value of s affect the convergence of a series?

The value of s directly affects the convergence of a series. If s is closer to 0, the series will converge more slowly and require more terms to reach a finite limit. On the other hand, if s is closer to 1, the series will converge more quickly and require fewer terms to reach a finite limit.

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