Does limit ln(n)/n^c -> 0 for any c>0?

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Homework Help Overview

The discussion revolves around the limit of the expression ln(n)/n^c as n approaches infinity, specifically questioning whether this limit approaches 0 for any positive constant c. The subject area is real analysis, focusing on limits and asymptotic behavior.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss potential proofs, including epsilon-delta arguments and BigO/SmallO notation. There is curiosity about the rigor expected in the class and the tools available for analysis. Questions are raised about the derivation of ln(n) and the application of the Bolzano-Weierstrass theorem.

Discussion Status

The discussion is active, with participants exploring different proof strategies and clarifying the context of the real analysis course. Some guidance has been offered regarding theorems that may be applicable, and there is an openness to further exploration of the limit.

Contextual Notes

Participants note that this is the first course in real analysis, which may influence the level of rigor and the types of proofs expected. There is also mention of the need to consider the behavior of the function in the real number context rather than just the natural numbers.

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Homework Statement



Does limit ln(n)/n^c -> 0 for any c>0?

Homework Equations





The Attempt at a Solution


I wonder if there is an
1.Epsilon Delta Proof
2.Proof using BigO SmallO notation.

Thanks
 
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It is obviously true. There are a lot of ways to prove this. What kind of class is this for? Are you expected to be rigorous in your solutions? N-epsilon proofs and asymptotic behaviour are very different arguments in terms of the development required in an analysis setting.
 
Yes, this is for real analysis class
 
What tools do you have available? How was ln(n) derived?

There is a nifty theorem that says that if a sequence converges then any subsequence will converge to the same limit. This will enable you to look at ln(x)/x^c in R rather than in N.
 
Yes, we can use Bolzano Weierstrasse Theorem. Please feel free to proceed.
My level is on baby Rudin, the this is the first course in real analysis.
 

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