# Thread: Analysis - complex sequences problem?

1. ## Analysis - complex sequences problem?

If you let {a_n} be a complex sequence where
lim [ |a_n|^(1/n) ] = q
i.e. the nth root of the modulus of the terms in the sequence tends to q.

(i) show that if q<1, then lim a_n=0
(ii) show that if q>1, then |a_n| --> infinity
iii) what can you say about the behaviour of |a_n| if q=1?

2. If $s_n$ is a real sequence and $\lim |s_n|^{1/n} = L$ then $s_n$ converges to zero if $L<1$.
So given a complex sequence we can write $a_n = x_n+iy_n$ thus $|a_n| = \sqrt{x_n^2+y_n^2}$.
$\lim |x_n|^{1/n} \leq \lim |\sqrt{x_n^2+y_n^2}|^{1/n} = L < 1 \implies \lim x_n \to 0$.
Similarly $\lim y_n=0$.