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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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 AP Posts: 137 Registered: 3/4/09
on real part of [(1+isqrt(7))/2]^n
Posted: Feb 10, 2014 12:00 PM

Be a=(1+isqrt(7))/2 and a_n=the real part of a^n

question : show lim |a_n | is +inf ? (it-is not a home-work..)

first values of 2a_n , n>=0, : 2/1/-3/-5/1/11/9/-13/-31/-5/57/

we have
(1-2z)/(2-z+z^2)=sum_{n>=0 } (a_{n+1}/2^n)z^n for |z|<sqrt(2)
=1/2-(3/4)z-(5/8)z^2+(1/16)z^3+...

(if b=(1-isqrt(7))/2, 1/(a-z)+1/(b-z)=...)

the radius of convergence is R>=sqrt(2)
but if R>sqrt(2) we obtain a contradiction because (1-2z)/(2-z+z^2) is
not define for a (|a|=sqrt(2))
so R=sqrt(2)

hence, if z=2 , the series diverges and |a_n| is not bounded.
But , after ...

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Date Subject Author
2/10/14 AP
2/10/14 Brian Q. Hutchings
2/10/14 Pubkeybreaker
2/10/14 Karl-Olav Nyberg
2/10/14 William Elliot
2/11/14 AP
2/11/14 Karl-Olav Nyberg
2/11/14 Robin Chapman
2/11/14 Karl-Olav Nyberg
2/11/14 Robin Chapman
2/11/14 Karl-Olav Nyberg
2/11/14 Karl-Olav Nyberg
2/15/14 Karl-Olav Nyberg
2/15/14 quasi
2/15/14 Karl-Olav Nyberg
2/15/14 quasi
2/11/14 William Elliot
2/11/14 Robin Chapman
2/11/14 William Elliot
2/13/14 quasi
2/14/14 Brian Q. Hutchings