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        riddles >> putnam exam (pure math) >> Rationalization
        
(Message started by: ThudanBlunder on Jul 18th, 2007, 4:56pm)
Title: Rationalization
Post by ThudanBlunder on Jul 18th, 2007, 4:56pm
Express
   1
---------  as a sum of terms of the form 2q, where q is rational.
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2 - 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2
Title: Re: Rationalization
Post by Barukh on Jul 19th, 2007, 11:16am
[hide]-.5 - n/6[/hide]?
Title: Re: Rationalization
Post by ThudanBlunder on Jul 19th, 2007, 11:50am

on 07/19/07 at 11:16:42, Barukh wrote:
[hide]-.5 - n/6[/hide]?

Pardon?
Title: Re: Rationalization
Post by Eigenray on Jul 19th, 2007, 12:08pm
Ah, [hide]geometric series[/hide].  And it's easy enough to turn that into a finite sum (which was probably intended):

[hide]2-3/6 + 2-4/6 + 2-5/6 + 2-6/6 + 2-7/6 + 2-8/6 + 2-9/6 + ...
= (2-3/6 + 2-4/6 + 2-5/6 + 2-6/6 + 2-7/6 + 2-8/6)(1+1/2+1/4+...)
= 23/6 + 22/6 + 21/6 + 1 + 2-1/6 + 2-2/6.[/hide]

This is much more elegant than my dull approach:

(a - b)(a - wb)(a - w2b)(-a - b)(-a - wb)(-a - w2b)
= (a3 - b3)(-a3 - b3)
= -a6 + b6
= -4,

where a = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif2, b=21/3, w=e2pi i/3.


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