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Alternating factorial series

  1. Apr 21, 2010 #1

    dba

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    1. The problem statement, all variables and given/known data
    Classify the series as absolute convergent, conditionally convergent, or divergent.

    [tex]
    \Sigma^{\infty}_{k = 1} (-1)^{k-1}\frac{k!}{(2k-1)!}
    [/tex]


    2. Relevant equations
    The Alternating Series Test: conditions for convergence
    decreasing
    lim --> infinity ak = 0


    3. The attempt at a solution
    I am not sure how to find out if the series is decreasing. Since it is a factorial, I cannot take the first deriative test.

    I was wondering if I should use the Alternating Series Test at all, since it is an factorial.

    Can someone help me here?
     
  2. jcsd
  3. Apr 21, 2010 #2

    lanedance

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    Homework Helper

    try the alternating series test

    a ratio of terms will show if the magnitude of terms decrease monotonically
     
  4. Apr 21, 2010 #3

    dba

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    Thanks.

    If I try the ratio I got stuck with the factorial.

    [tex]
    \frac{(k+1)!}{(2(k+1)-1)!} * \frac{(2k-1)!}{k!} = \frac{(2k-1)!}{(2(k+1)-1)!} * (k+1)[/tex]


    Can I write

    [tex](2(k+1)-1)! = (2k+2-1)! = (2k+1)![/tex]
     
  5. Apr 21, 2010 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Yes, of course. And so you have (2k-1)!/(2k+1)!= (2k-1)!/[(2k+1)(2k)(2k-1)!]
     
  6. Apr 22, 2010 #5

    dba

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    Oh, ok.
    Thank you! I was able to solve this one :smile:
     
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