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July 20th, 2019, 08:45 AM   #1
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Prove inequality

(1) Prove that $\displaystyle \frac{\tan(x)}{x}<2-\sqrt{1-x^{2}}\; $,for $\displaystyle 0<x\leq 1$.

(2) Prove that $\displaystyle \frac{|\sin(N)|}{N} +\frac{|\sin(N+1)|}{N+1} +...+\frac{|\sin(2N)|}{2N}>\frac{1}{6} \; $ for $\displaystyle N\in \mathbb{N}$.

Last edited by skipjack; July 20th, 2019 at 01:44 PM.
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July 20th, 2019, 08:59 AM   #2
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is this a brain teaser or do you need help?
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July 20th, 2019, 10:13 AM   #3
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Quote:
Originally Posted by romsek View Post
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.
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July 23rd, 2019, 03:24 PM   #4
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… working them for fun.
Please show your work, so far.

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July 24th, 2019, 06:03 AM   #5
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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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