August 12th, 2019, 05:46 AM  #1 
Senior Member Joined: Dec 2015 From: somewhere Posts: 592 Thanks: 87  Solve inequality
For which values of N : $\displaystyle \; \sum_{j=1}^{N} j^{1} >\frac{N}{\ln(N+1)}$.
Last edited by idontknow; August 12th, 2019 at 05:51 AM. 
August 12th, 2019, 06:25 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,921 Thanks: 2203 
Do you know of any natural number for which that inequality doesn't hold?

August 12th, 2019, 08:39 AM  #3  
Senior Member Joined: Dec 2015 From: somewhere Posts: 592 Thanks: 87  Quote:
The right side is increasing for x>e and the rate of divergence is larger than the left side so $\displaystyle N\geq \lceil e \rceil =3$. If the left side is smaller than right side for N>2 then it is larger for N<3 .  
August 12th, 2019, 04:23 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,921 Thanks: 2203  

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