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Math Help - Complex convergence

  1. #1
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    Complex convergence

    Hey,

    At first I thought part a and c both converged but now after reading that "A sequence converges precisely when both the Im(z) and the Re(z) converge" which in this set of questions makes each of them converge. Is that correct?

    Part b seems to repeat after 6ish n's and continues to repeat for large n which makes me think it doesn't converge.

    Also would the limit for a and c be zero? as cos/sin are between -1 and 1 and (c) because sqrt(3)^n > sqrt(2) which is the norm of (1+i)

    Thanks in Advance,

    Daniel
    Last edited by Daniiel; August 30th 2011 at 10:25 PM.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Complex convergence

    (a) For z_n=\frac{\cos n\theta+i\sin n\theta}{n} , real and imaginary parts converge to 0 so, ...

    (b) For z_n=\frac{(1+i)^n}{(\sqrt{2})^n}=\frac{(\sqrt{2})^  n(\cos (n\pi/4)+i\sin (n\pi/4))}{(\sqrt{2})^n}=\cos (n\pi/4)+i\sin (n\pi/4) so ...

    (c) For z_n=\frac{(1+i)^n}{(\sqrt{3})^n} , |z_n|\to 0 so ...
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  3. #3
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    Re: Complex convergence

    Thanks Fernando,

    So just to double check,

    I was right to think b diverges because e^(i n pi/4) ->inifnity as n -> infinity
    Last edited by Daniiel; August 30th 2011 at 10:43 PM.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: Complex convergence

    Quote Originally Posted by Daniiel View Post
    I was right to think b diverges because e^(i n pi/4) ->inifnity as n -> infinity
    No, it is an oscillating sequence.
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