
Re: on real part of [(1+isqrt(7))/2]^n
Posted:
Feb 11, 2014 11:25 PM


> 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,[(n1)/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

