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Math Help - Analysis - complex sequences problem?

  1. #1
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    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?

    please help!
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  2. #2
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    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}.
    So,
    \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.
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