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 January 19th, 2019, 04:14 AM #1 Senior Member   Joined: Dec 2015 From: iPhone Posts: 436 Thanks: 68 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 05:04 AM.
 January 19th, 2019, 05:09 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,373 Thanks: 2010 As $n$ increases, $H_n$ increases and $3/n$ decreases. Thanks from topsquark and idontknow
 January 19th, 2019, 09:52 AM #3 Senior Member   Joined: Dec 2015 From: iPhone Posts: 436 Thanks: 68 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 03:33 PM.
 January 19th, 2019, 02:19 PM #4 Global Moderator   Joined: May 2007 Posts: 6,704 Thanks: 670 Note that the inequality holds only for $n\ge 3$. Thanks from idontknow
 January 19th, 2019, 02:43 PM #5 Senior Member   Joined: Dec 2015 From: iPhone Posts: 436 Thanks: 68 I meant inequation; I wrote it wrong. Not inequality but inequation. Last edited by skipjack; January 19th, 2019 at 03:34 PM.
January 20th, 2019, 12:56 PM   #6
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Quote:
 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, 01:00 PM #7 Senior Member   Joined: Dec 2015 From: iPhone Posts: 436 Thanks: 68 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, 01:54 PM #8 Global Moderator   Joined: Dec 2006 Posts: 20,373 Thanks: 2010 Are you sure you mean n/3 rather than 3/n?
 January 20th, 2019, 02:32 PM #9 Senior Member   Joined: Dec 2015 From: iPhone Posts: 436 Thanks: 68 3/n .

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