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

[grossgermany]

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

 

[betel]

What about taking derivatives and comparing them?

 

[Susanne217]

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

$$ln(n) < n^c$$

 

[grossgermany]

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