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 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? Thanks from topsquark and idontknow
August 12th, 2019, 08:39 AM   #3
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Quote:
 Originally Posted by idontknow For which values of N : $\displaystyle \; \sum_{j=1}^{N} j^{-1} >\frac{N}{\ln(N+1)}$.
In short-terms by calculus :
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
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Quote:
 Originally Posted by idontknow The right side is increasing for x>e
There is no $x$ in the inequality.

Quote:
 Originally Posted by idontknow . . . the rate of divergence
What do you mean by "rate of divergence"?

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