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Complex Variables

  1. Sep 25, 2007 #1
    I can't think of how to title the problem I'm having, but this is what the course is called. Complex being imaginary numbers, ie z = a + ic where i is the sqrt of -1.

    So here is the question that I have no idea where to start with:

    Construct a sequence {zn} which is bounded and for which the successive
    terms get increasingly closer, but which is not convergent. In other words,
    {zn} must satisfy:
    (i) For some B > 0, |zn| < B for every n = 1, 2,...
    (ii) For every n, |zn+2 - zn+1| < |zn+1 - zn|.
    (iii) {zn} diverges.
    Note that the inequality in (ii) is strict. Make sure to prove that your
    sequence satisfies all three parts.

    n is a subscript of z.

    Thanks in advance.
  2. jcsd
  3. Sep 25, 2007 #2


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    Let tn=sum(1/i) for i=1 to n. Consider zn=exp(i*tn). Now you just have to do the proofs.
  4. Sep 26, 2007 #3
    Thanks. On the same homework assignment, I have another problem but this is more with the question itself. What does this mean: limit of Arg z, as z approaches zero?
  5. Sep 26, 2007 #4


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    ?? Just what it says! Or is the question rather "what does Arg z mean"?

    Any complex number z= x+ iy can be written in polar form: [itex]z= r (cos(\theta)+ i sin(\theta)[/itex] or simply as [itex]z= r e^{i\theta}[/itex]. In either case [itex]Arg z= \theta[/itex].

    Take a look at z= x+ ix. What is Arg z for all x? What is its limit as z (and so x) goes to 0?
    Now look at z= x- ix. What is Arg z for all x? What is its limit as z (and so x) goes to 0?
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