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May 16th, 2019, 05:37 AM   #1
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Range of N

Find the range of N such that $\displaystyle 1+\frac{1}{2}+...+\frac{1}{N} >(1+\frac{1}{N})^{N} $ .
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May 16th, 2019, 07:22 AM   #2
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Apparently, $N > 6$.
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May 17th, 2019, 04:33 AM   #3
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To save time and use less complicated symbols I will post the solution in short-terms :
Using (*)$\displaystyle H_{2^{N}} \geq 1+\frac{N}{2}$ ; $\displaystyle (1+1/N)^{N}$ is close(bounded) to 2 and 3 for integers .
Let m be the mean value of number of terms of $\displaystyle H_{N}$ with condition(*) .
For the bound close to 2 , $\displaystyle N=4$ ; For the bound close to 3 , $\displaystyle N=8$.
Now $\displaystyle m=\lfloor \frac{b_1 + b_2 }{2} \rfloor =\lfloor \frac{8+4}{2}\rfloor=12/2=6$ , so $\displaystyle N>6$.
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Last edited by idontknow; May 17th, 2019 at 04:44 AM.
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