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Homework Help: 2 Complex Series problems

  1. Nov 18, 2012 #1
    1. The problem statement, all variables and given/known data

    pardon my terrible latex skills

    Find the limit of this series:

    [itex]\sum[/itex] (n = 0 to ∞) (-1)[itex]^{n}[/itex]([itex]\frac{2}{3}[/itex])[itex]^{n}[/itex]

    2. Relevant equations

    No idea, it looks like an alternating series test, but I am supposed to actually find the sum, not just whether or not it converges.

    3. The attempt at a solution

    No idea


    1. The problem statement, all variables and given/known data

    [itex]\sum[/itex](k = 1 to ∞) [itex]\frac{(-1)^{k}k^{3}}{(1+i)^{k}}[/itex]

    2. Relevant equations




    3. The attempt at a solution

    Once again, it looks like an alternating series. I tried the root test and got (1/2)-(1/2)i, but then I realized the root test is not applicable because the series is complex. No way to compare (1/2) - (1/2)i to the real number 1 in terms of ordering. Its not geometric so I don't have a formula for finding the sum.
     
    Last edited: Nov 18, 2012
  2. jcsd
  3. Nov 18, 2012 #2
    Found the answer to the first problem,

    Its a geometric series if you rewrite as Ʃ (-2/3)^n

    Ratio is (-2/3) so the limit is 1/(1+2/3) = 3/5

    I still have no idea how to do the second problem
     
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