general form of the old ineq

romano · Riemann Hypothesis Offline Joined: 10 Jun 2004 Posts: 271

Let $t \geq 0$ , $x_n = \frac{1+t+t^2+..+t^n}{n+1}$ .
Prove that : $^{n-1}\sqrt{x_{n-1}} \leq ^n\sqrt{x_n}$ for all $n \geq 1$

**Posted:** Sun Jan 30, 2005 9:19 am

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blahblahblah

It doesn't hold for $n=1$ ;

$1\leq \frac {1+t}{2}$ is not generally true. I think we at least need $t\geq 1$ .

**Posted:** Sun Jan 30, 2005 11:13 pm

Myth

No we no need it, since inequalities for $t$ and $1/t$ are the same Wink

Forget about $n=1$ Smile

**Posted:** Sun Jan 30, 2005 11:48 pm

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romano · Riemann Hypothesis Offline Joined: 10 Jun 2004 Posts: 271

Anybody , let's try to solve it !

**Posted:** Mon Jan 31, 2005 9:21 am

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zhaobin

**Posted:** Mon Jan 31, 2005 9:41 am

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Myth

No, I am right Wink

**Posted:** Mon Jan 31, 2005 9:43 am

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Myth is out of here

zhaobin

yes I'm sorry

**Posted:** Mon Jan 31, 2005 9:46 am

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I am a college student at a not well known universty. I am interested in math. I believe that one coin has two sides.

romano · Riemann Hypothesis Offline Joined: 10 Jun 2004 Posts: 271

For more understoodable explanation : We need to show that :
$x_1 \leq \sqrt{x_2} \leq ^3\sqrt{x_3} \leq ^4{x_4} \leq ....$

**Posted:** Mon Jan 31, 2005 9:50 am

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zhaobin

assume t>1 can we let $f(t)=\frac{(t^{n+1}-1)^{n-1}(t-1)}{(t^{n}-1)^n}$
can we prove that $f'(t)>0$ ?

**Posted:** Mon Jan 31, 2005 12:28 pm

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I am a college student at a not well known universty. I am interested in math. I believe that one coin has two sides.

romano · Riemann Hypothesis Offline Joined: 10 Jun 2004 Posts: 271

Ican't prove $f'(t) > 0$
How can you do with $0 < t < 1$
In short , this problem is still unsolved ! Sad

**Posted:** Tue Feb 01, 2005 9:40 am

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