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Topic: on real part of [(1+isqrt(7))/2]^n
Replies: 20   Last Post: Feb 15, 2014 5:52 AM

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Brian Q. Hutchings

Posts: 6,017
Registered: 12/6/04
Re: on real part of [(1+isqrt(7))/2]^n
Posted: Feb 10, 2014 2:04 PM
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is that a continued fraction

look, you can simplify the complex number,
using properties of i

> first values of 2a_n , n>=0, : 2/1/-3/-5/1/11/9/-13/-31/-5/57/
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> we have
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> (1-2z)/(2-z+z^2)=sum_{n>=0 } (a_{n+1}/2^n)z^n for |z|<sqrt(2)
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> =1/2-(3/4)z-(5/8)z^2+(1/16)z^3+...
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> (if b=(1-isqrt(7))/2, 1/(a-z)+1/(b-z)=...)
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> the radius of convergence is R>=sqrt(2)
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> but if R>sqrt(2) we obtain a contradiction because (1-2z)/(2-z+z^2) is
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> not define for a (|a|=sqrt(2))
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> so R=sqrt(2)
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> hence, if z=2 , the series diverges and |a_n| is not bounded.
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> But , after ...
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> ---
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> Ce courrier électronique ne contient aucun virus ou logiciel malveillant parce que la protection avast! Antivirus est active.
>
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