Is ln(n) Less Than n^c for All c>0 and n>N?

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Homework Help Overview

The discussion revolves around proving the inequality ln(n) < n^c for all real c > 0 and for sufficiently large n, specifically n > N. This falls within the realm of real analysis and involves comparing the growth rates of logarithmic and polynomial functions.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants suggest various methods for proving the inequality, including graphical analysis and derivative comparisons. There is also mention of using series and the comparison test from calculus to establish the relationship between ln(n) and n^c.

Discussion Status

The discussion is ongoing, with participants exploring different approaches and raising questions about the validity of their methods. Some guidance has been offered regarding the comparison test, but no consensus has been reached on a definitive proof.

Contextual Notes

Participants are considering the need for a specific constant M in the comparison test and the implications of establishing N for the inequality to hold. There is an emphasis on rigor in the proof, consistent with the requirements of real analysis.

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Homework Statement



How to rigorously (real analysis) prove that for all real c>0
Exists N such that for all n>N
ln(n)<n^c

Homework Equations





The Attempt at a Solution


The fact can be shown using graphical calculator
 
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What about taking derivatives and comparing them?
 
grossgermany said:

Homework Statement



How to rigorously (real analysis) prove that for all real c>0
Exists N such that for all n>N
ln(n)<n^c

Homework Equations





The Attempt at a Solution


The fact can be shown using graphical calculator

I think that you can do it like this:

If you view these two function as series

e.g. \sum_{n=1}^{\infty} ln(n) and \sum_{c=1}^{\infty} n^c and then use the comparison test from Calculus to show that

ln(n) &lt; n^c
 
For the comparison test, we need to show that there exists N such that for all n>N
ln(n)<Mn^c for some constant M
 

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