Proofing Convergence of a_n = n^{1/n} to 1 - Help Needed!

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In summary, the conversation discusses the proof of the limit of a sequence, a_n = n^{1/n} \to 1, without the use of the exponential function. The participants suggest using ln or l'Hopital's rule, but the person asking for help is hesitant to use them as they were not introduced in their course yet. It is suggested to seek assistance from a math assistant for a different approach.
  • #1
nonequilibrium
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Hello, I was wondering how to proof [tex]a_n = n^{1/n} \to 1.[/tex]

Doing it straight from the definition got me nowhere. But I was thinking. It is obvious that [tex]\liminf a_n \geq 1[/tex] (since otherwise for big n you could get n^(1/n) < 1 <=> n < 1). And I also have already a proof of [tex]\limsup x_n^{1/n} \leq \limsup \frac{x_{n+1}}{x_n}[/tex] (which is a general result for any row x_n).

But proving [tex]\limsup \frac{a_{n+1}}{a_n} \leq 1[/tex] seemed to be harder than I thought.

So I'm completely stuck. Any ideas?

Thank you,
mr. vodka
 
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  • #2
I don't think you need to use lim sup/inf here.

n1/n=eln(n1/n)=eln(n)/n.

So you just need to show that ln(n)/n-->0. Try using l'Hopital's rule.
 
  • #3
Hm, I'd like to proof it without the use of the exponential function. It's namely introduced in my Analysis course before the exp function, and it's actually used in a proof about power series, which is later used to introduce the e-function. Thanks for your help!
 
  • #4
If you don't want to use exp function, how about using ln (natural log)?
ln(an) = ln(n)/n -> 0.
 
  • #5
Well, we defined that as the inverse of e. Any possibilities without e (or ln)? I appreciate the help though! I find the ln/e proofs very elegant, but I hope you understand I'm going to choose for logical consistency in my course :)
 
  • #6
I'm not sure that there's a way around that doesn't use exp or ln.
 
  • #7
Oh... But it was left as an exercise for us in our course, so there must be. Hmmm, maybe I should contact one of the math assistants for this one then. Thank you guys for your time.
 
  • #8
For those that are interested: http://myyn.org/m/article/limit-of-nth-root-of-n/
 
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FAQ: Proofing Convergence of a_n = n^{1/n} to 1 - Help Needed!

How do you prove the convergence of a sequence?

To prove the convergence of a sequence, you need to show that the terms of the sequence get closer and closer to a specific value, called the limit, as the sequence progresses. This can be done by using various mathematical techniques such as the squeeze theorem, the ratio test, and the root test.

What is the sequence a_n = n^{1/n}?

The sequence a_n = n^{1/n} is a sequence of numbers where each term is equal to the nth root of n. This means that as n increases, the terms of the sequence get closer and closer to 1.

How do you prove that a sequence converges to a specific limit?

To prove that a sequence converges to a specific limit, you need to show that for any given small number, there exists a corresponding term in the sequence after which all the terms are within that small number of the limit. This can be done through the use of mathematical proofs and techniques.

Can you explain the concept of convergence in simpler terms?

Convergence in simpler terms means that the terms of a sequence get closer and closer to a specific value as the sequence progresses. This value is called the limit. It is similar to a target that the terms of the sequence are trying to hit.

How can I determine the limit of a sequence?

The limit of a sequence can be determined by evaluating the terms of the sequence as n approaches infinity. If the terms of the sequence approach a specific value, then that value is the limit of the sequence. However, in some cases, the limit may not exist or may be infinite, which can be determined by using mathematical techniques such as the ratio test or the root test.

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