My Math Forum Prove inequality

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 July 20th, 2019, 08:45 AM #1 Senior Member   Joined: Dec 2015 From: somewhere Posts: 592 Thanks: 87 Prove inequality (1) Prove that $\displaystyle \frac{\tan(x)}{x}<2-\sqrt{1-x^{2}}\;$,for $\displaystyle 0\frac{1}{6} \;$ for $\displaystyle N\in \mathbb{N}$. Last edited by skipjack; July 20th, 2019 at 01:44 PM.
 July 20th, 2019, 08:59 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,529 Thanks: 1389 is this a brain teaser or do you need help? Thanks from idontknow
July 20th, 2019, 10:13 AM   #3
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Quote:
 Originally Posted by romsek is this a brain teaser or do you need help?
It’s not for help; I’m just working them for fun.

Last edited by skipjack; July 20th, 2019 at 01:48 PM.

July 23rd, 2019, 03:24 PM   #4
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Quote:
 Originally Posted by idontknow … working them for fun.

 July 24th, 2019, 06:03 AM #5 Senior Member   Joined: Dec 2015 From: somewhere Posts: 592 Thanks: 87 Yes, about inequality (2): Since $\displaystyle S_{n}=\sum \frac{|\sin(n)|}{n}$ is increasing, $\displaystyle N=1$ gives the minimal number of terms, which is $\displaystyle 2$. By the first two terms: $\displaystyle \frac{|\sin(N)|}{N}+\frac{|\sin(1+N)|}{1+N}>\frac{ 1}{6}$ remains to prove the whole inequality. Can we somehow continue? Last edited by skipjack; July 24th, 2019 at 10:45 PM.

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