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November 4th, 2019, 04:56 AM   #1
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Range of N

Find the range of $\displaystyle N\in \mathbb{N}$ such that $\displaystyle H_N > (1+\frac{1}{N^2 })^{N}$.
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November 4th, 2019, 01:28 PM   #2
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What is $H_N$?
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November 5th, 2019, 01:04 AM   #3
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Originally Posted by idontknow View Post
Find the range of $\displaystyle N\in \mathbb{N}$ such that $\displaystyle H_N > (1+\frac{1}{N^2 })^{N}$.
Find the range of $\displaystyle N\in \mathbb{N}$ such that $\displaystyle H_{N^2 }=1+1/2+1/3+\dotsc +1/N^{2} > (1+\frac{1}{N^2 })^{N}$.
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November 5th, 2019, 02:23 PM   #4
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The series is increasing with $N$ while the r.h.s. is decreasing with $N$. Graph them and see where they cross.
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November 6th, 2019, 02:05 AM   #5
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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.
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November 6th, 2019, 08:02 AM   #6
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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.
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November 6th, 2019, 09:10 AM   #7
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I agree , no more brain teasers .
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