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Alternating Series

  1. Sep 19, 2010 #1
    Consider the series (to infinity; n=0) (-1/3)^n (x-2)^n

    Find the intercal of convergence for this series.

    To what function does this series converage over this interval?

    I know this is an alternating series...I just don't know how to go about it. Thanks for you help.

    ** Should I do this like a partial sum? Do I just keep multiplying them together until they reach Zero *because then it has converaged? **
    Last edited: Sep 19, 2010
  2. jcsd
  3. Sep 19, 2010 #2


    Staff: Mentor

    It might be helpful to write this series as
    [tex]\sum_{n = 0}^{\infty} (-1)^n \left(\frac{x - 2}{3}\right)^n[/tex]

    For some values of x, this is an alternating series, but for others, it's not.
    What theorems do you know for determining whether a series converges?
  4. Sep 19, 2010 #3
    a [tex]_{n+1}[/tex][tex]\leq[/tex] for all n

    lim[tex]_{n\rightarrow\infty}[/tex] a[tex]_{n}[/tex] = 0

    **sorry those are supposed to be lower subscripts**
  5. Sep 19, 2010 #4


    Staff: Mentor

    You're still thinking that this is an alternating series. For some values of x (such as x = 0), it's NOT an alternating series.

    Do you know any tests other than the alternating series test?
    Last edited: Sep 19, 2010
  6. Sep 19, 2010 #5
    Ratio Test so...lim |a [tex]_{n+1}[/tex]| / |a[tex]_{n}[/tex]|
  7. Sep 20, 2010 #6


    Staff: Mentor

    OK, so what do you get if you use the Ratio Test?
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