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 May 16th, 2019, 05:37 AM #1 Senior Member   Joined: Dec 2015 From: somewhere Posts: 591 Thanks: 87 Range of N Find the range of N such that $\displaystyle 1+\frac{1}{2}+...+\frac{1}{N} >(1+\frac{1}{N})^{N}$ . May 16th, 2019, 07:22 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,919 Thanks: 2203 Apparently, $N > 6$. Thanks from idontknow May 17th, 2019, 04:33 AM #3 Senior Member   Joined: Dec 2015 From: somewhere Posts: 591 Thanks: 87 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 $\displaystyle N>6$. Thanks from topsquark Last edited by idontknow; May 17th, 2019 at 04:44 AM. Tags range Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post zylo Complex Analysis 13 December 4th, 2017 05:47 PM ahmed93 Calculus 7 November 4th, 2012 12:03 PM tale Computer Science 5 May 23rd, 2012 05:16 AM panky Calculus 0 March 20th, 2012 05:30 AM panky Calculus 3 January 20th, 2012 04:57 AM

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