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 January 19th, 2019, 05:14 AM #1 Senior Member   Joined: Dec 2015 From: Earth Posts: 828 Thanks: 113 Math Focus: Elementary Math Solve inequality Which inequality of the harmonic series should I use to solve the inequality? $\displaystyle H_n > \frac{3}{n}\; \;$ $\displaystyle H_n =1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$ Last edited by skipjack; January 19th, 2019 at 06:04 AM. January 19th, 2019, 06:09 AM #2 Global Moderator   Joined: Dec 2006 Posts: 21,110 Thanks: 2326 As $n$ increases, $H_n$ increases and $3/n$ decreases. Thanks from topsquark and idontknow January 19th, 2019, 10:52 AM #3 Senior Member   Joined: Dec 2015 From: Earth Posts: 828 Thanks: 113 Math Focus: Elementary Math It is enough to write $\displaystyle \frac{3}{n}=\frac{1}{n}+\frac{1}{n}+\frac{1}{n}$ Write $\displaystyle H_n$ as $\displaystyle \frac{1}{n}+\frac{1}{n-1} +\frac{1}{n-2}+...+1$$\displaystyle >\frac{3}{n}$ The first 3 terms of $\displaystyle H_n$ are larger than $\displaystyle \frac{3}{n}\; \;$, which means the number of terms must be 3 or more (because the number of terms of the H is equal to $\displaystyle n$). $\displaystyle n\geq 3$ Thanks from topsquark Last edited by skipjack; January 19th, 2019 at 04:33 PM. January 19th, 2019, 03:19 PM #4 Global Moderator   Joined: May 2007 Posts: 6,852 Thanks: 743 Note that the inequality holds only for $n\ge 3$. Thanks from idontknow January 19th, 2019, 03:43 PM #5 Senior Member   Joined: Dec 2015 From: Earth Posts: 828 Thanks: 113 Math Focus: Elementary Math I meant inequation; I wrote it wrong. Not inequality but inequation. Last edited by skipjack; January 19th, 2019 at 04:34 PM. January 20th, 2019, 01:56 PM   #6
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 Originally Posted by idontknow I meant inequation; I wrote it wrong. Not inequality but inequation.
What do you mean by "inequation"? I don't believe there is such a word. January 20th, 2019, 02:00 PM #7 Senior Member   Joined: Dec 2015 From: Earth Posts: 828 Thanks: 113 Math Focus: Elementary Math I mean: find interval of $\displaystyle n$ or the range of $\displaystyle n$ Such that inequality $\displaystyle H_n \geq \frac{n}{3}$ holds true. January 20th, 2019, 02:54 PM #8 Global Moderator   Joined: Dec 2006 Posts: 21,110 Thanks: 2326 Are you sure you mean n/3 rather than 3/n? January 20th, 2019, 03:32 PM #9 Senior Member   Joined: Dec 2015 From: Earth Posts: 828 Thanks: 113 Math Focus: Elementary Math 3/n . Tags inequality, solve 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 DerGiLLster Algebra 2 September 6th, 2015 03:03 PM mattematicarka Algebra 2 February 6th, 2013 12:22 PM Punch Algebra 3 July 2nd, 2012 01:43 PM jatt-rockz Algebra 5 November 10th, 2011 06:07 AM xx3004 Algebra 8 December 22nd, 2010 10:00 PM

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