November 4th, 2019, 04:56 AM  #1 
Senior Member Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math  Range of N
Find the range of $\displaystyle N\in \mathbb{N}$ such that $\displaystyle H_N > (1+\frac{1}{N^2 })^{N}$.

November 4th, 2019, 01:28 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,852 Thanks: 743 
What is $H_N$?

November 5th, 2019, 01:04 AM  #3 
Senior Member Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math  
November 5th, 2019, 02:23 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,852 Thanks: 743 
The series is increasing with $N$ while the r.h.s. is decreasing with $N$. Graph them and see where they cross.

November 6th, 2019, 02:05 AM  #5 
Senior Member Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math 
The RHS has range $\displaystyle (1,e)$ , the integer inside is $\displaystyle 2$. Applying $\displaystyle H(2^{k} )\geq 1+k/2=2 \; \Rightarrow k=2$ and $\displaystyle N^2 =2^{k}=4\; $ or $\displaystyle N\geq 2$. Last edited by idontknow; November 6th, 2019 at 02:09 AM. 
November 6th, 2019, 08:02 AM  #6 
Senior Member Joined: Sep 2015 From: USA Posts: 2,638 Thanks: 1473 
These are all brain teasers right? You don't need these questions answered for class or anything do you? If so I would ask that you put them in the "Math" forum with a label indicating that they are brain teasers for fun. This way helpers with limited time (and isn't that all of us?) can focus on helping students with their problems first. Thank you. 
November 6th, 2019, 09:10 AM  #7 
Senior Member Joined: Dec 2015 From: Earth Posts: 823 Thanks: 113 Math Focus: Elementary Math 
I agree , no more brain teasers . 

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