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        riddles >> putnam exam (pure math) >> Inequality
        
(Message started by: THUDandBLUNDER on Aug 15th, 2006, 8:58pm)
Title: Inequality
Post by THUDandBLUNDER on Aug 15th, 2006, 8:58pm

on 08/15/06 at 17:13:47, SWF wrote:
By the way, we seemed to hurting for new riddles around here so I was just trying to supply one.

Prove that for all n sqrt(2sqrt(3sqrt(4sqrt(5..........sqrt(n).........)))) < 3
Title: Re: Inequality
Post by Eigenray on Aug 15th, 2006, 11:51pm
[hideb]Taking log=log2, we are to show

L = log(2)/2 + log(3)/22 + log(4)/23 + log(5)/24 + log(6)/25 + ... < log 3.

Now,
log(5)/24 + log(6)/25 + ... < log(8)(1/24+...+1/27) + log(16)(1/28+...+1/215) + log(32)(1/216+...+1/231) + ...
= 3(1/24+...) + (4-3)(1/28+...) + (5-4)(1/216+...) + ...
= 3/23 + 1/27 + 1/215 + ...
< 3/8 + 1/64.

That is,
L < log(2)/2 + log(3)/4 + log(4)/8 + (3/8+1/64) = 1 + log(3)/4 + 1/8 + 1/64.

To show L < log(3), it suffices to note
24L < 243 21/221/16
< 16 3 (3/2) (1+1/9) = 80 < 81.[/hideb]
Title: Re: Inequality
Post by srn347 on Sep 23rd, 2007, 9:26pm
I would disprove it, but my scientific calculator gets stack error after 13 or something around there.
Title: Re: Inequality
Post by ima1trkpny on Sep 23rd, 2007, 9:45pm

on 09/23/07 at 21:26:49, srn347 wrote:
I would disprove it, but my scientific calculator gets stack error after 13 or something around there.

LOL... update to a decent graphing/programable calculator... it will save you quite a bit of trouble.
Title: Re: Inequality
Post by srn347 on Sep 23rd, 2007, 10:30pm
it is decent. And also, it's not software, it's an actual calculator, so it has finite memory.
Title: Re: Inequality
Post by mikedagr8 on Sep 23rd, 2007, 11:53pm
Why not just use pen and paper? ::)
Title: Re: Inequality
Post by ima1trkpny on Sep 24th, 2007, 1:05am

on 09/23/07 at 23:53:15, mikedagr8 wrote:
Why not just use pen and paper? ::)

I would have suggested that also... but I have a strong feeling he can't do it without the help of a calculator... so I was just suggesting he use the best tool available that he is capable of understanding.  ::) And LOL, the beauty of programable graphing calculators (since he's in school) is that you can program common formulas in so all you have to do is give it the variables. (However I would recommend knowing how to do these things without a calculator first.)
Title: Re: Inequality
Post by mikedagr8 on Sep 24th, 2007, 1:16am

on 09/24/07 at 01:05:36, ima1trkpny wrote:
I would have suggested that also... but I have a strong feeling he can't do it without the help of a calculator... so I was just suggesting he use the best tool available that he is capable of understanding.  ::) And LOL, the beauty of programable graphing calculators (since he's in school) is that you can program common formulas in so all you have to do is give it the variables. (However I would recommend knowing how to do these things without a calculator first.)


Hey, I have a programmable calculator. I only use it for a few equations, but these are equations which take me a full page or more of working out.

Anything I need to know I memorise, well as much as I can. And you are correct, there is no point in using something on a calculator unless you know how to do it by hand.
On tests I have lost too many marks because I was lazy and just reffered to the calculator instead of showing my working out. E.g. Simultaneuos equations.
Title: Re: Inequality
Post by srn347 on Sep 24th, 2007, 6:23pm
Pen and paper for stacking roots?! That's crazy! Anyway, square both sides until the left side is rational. 3^(2^n)>n!


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