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

(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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