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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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William Elliot

Posts: 2,637
Registered: 1/8/12
Re: on real part of [(1+isqrt(7))/2]^n
Posted: Feb 11, 2014 11:25 PM
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> Let a=(1+isqrt(7))/2 and a_n=the real part of a^n
> question : show lim |a_n | is +inf ?

That's not a question. That's an exercise.

Let r = sqr 7, b = 1 + ir.

Below I show how to prove the limit is infinite for 1 + i.sqr 7.
The same proof will suffice for your problem.

b^n = sum(j=0,n) n_C_j (ir)^j
= sum(j=0,[n/2]) n_C_2j (-7)^j + i.sum(j=0,[(n-1)/2]) n_C_(2j+1)

re(b^n) = sum(j=0,[n/2]) n_C_2j (-7)^j

d_n = sum(j=1,[n/2]) n_C_2j (-7)^j < re(b^n)
(d_n)^2 = sum(j,k=1,[n/2]) n_C_2j n_C_2k 7^(j+k) -> oo as n->oo.

Thus |d_n| and |b_n| -> oo.
Divide through appropriately by 2^n to show |a_n| -> oo

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