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How do you test the convergence of i^n (imaginary number) or (-1)^n?

  1. Sep 14, 2005 #1
    I was thinking alternating series test, but cant that only be used for testing for convergence, not for testing divergence?
     
  2. jcsd
  3. Sep 14, 2005 #2

    Hurkyl

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    How about the divergence test? :tongue2:
     
  4. Sep 14, 2005 #3

    LeonhardEuler

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    You'll have to be more specific. The convergence of what? The sequence of terms, [itex] a_n=i^n[/itex], the series, [itex]\sum_{n=0}^{\infty}i^n[/itex], or some sequence or series that just has [itex]i^n[/itex] as one term? I can tell you right now, the first two diverge.
     
  5. Sep 14, 2005 #4
    The divergence test doesnt work for indeterminant forms.
     
  6. Sep 14, 2005 #5

    I must prove that [itex]\sum_{n=0}^{\infty}i^n[/itex] diverges.

    Or rather, I must prove wheter or not its convergent of divergent. I know its divergent, but I still have to prove it.
     
  7. Sep 14, 2005 #6

    LeonhardEuler

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    [itex]i^n[/itex] is not an indeterminant form.
     
  8. Sep 14, 2005 #7

    Hurkyl

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    There isn't an indeterminant form involved. :tongue2:
     
  9. Sep 14, 2005 #8

    LeonhardEuler

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    Oh, that's simple. Use the divergence test as Hurkyl suggested. Do the terms in the series approach zero as n approaches infinty?
     
  10. Sep 14, 2005 #9
    The limit of [itex]i^n[/itex] as n goes to infinity doesnt have a solution does it?
     
  11. Sep 14, 2005 #10

    LeonhardEuler

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    Right, exactly!
     
  12. Sep 14, 2005 #11
    No, they alternate ... i,-1,-i,1,i,-1,...
     
  13. Sep 14, 2005 #12
    But that is not the same as saying indeterminant form?
     
  14. Sep 14, 2005 #13

    LeonhardEuler

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    So the limit doesn't exist. Then can it equal zero?
     
  15. Sep 14, 2005 #14

    LeonhardEuler

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    No, an indeterminant form is something like [itex]\frac{0}{0}[/itex], or [itex]\frac{\infty}{\infty}[/itex], or [itex]1^{\infty}[/itex]. A non-existant limit is just a non-existant limit.
     
  16. Sep 14, 2005 #15

    Hurkyl

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    An indeterminate form is a "limit form" for which we don't have enough information to say what the limit really is, or if it exists. For example, [itex]\lim_{x \rightarrow 0} x/x[/itex] has the indeterminate form 0/0.
     
  17. Sep 14, 2005 #16
    Its usually always silly mistakes :D
    Thanks alot guys..
     
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