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Homework Help: Raabe's criteria

  1. Mar 15, 2006 #1

    MathematicalPhysicist

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    i'm having a problem proving raabe's criteria for convergent sums.
    here in planetmath there's a description of it:
    http://planetmath.org/encyclopedia/RaabesCriteria.html

    i got that the first inequality is correct when mu is smaller than 1.

    i got a hint in my test that i should show that {na_(n+1)} is monotonely decreasing, which i did and by another theorem to show c_n=(n-1)a_n-na_(n+1) is convergent which i also did, but i got that
    (1-a_(n+1)/a_n)*n=(1-(1-1/n))*n=1>=mu.
    where have i gone wrong?

    thanks in advance.
     
    Last edited by a moderator: Apr 22, 2017
  2. jcsd
  3. Mar 15, 2006 #2
    a Mathematical series is shown in this fasion:
    a_(n+1) = a_n * q
    if |q|<1 then the series is convergent.

    in your case, you see than q = a_(n+1)/a_n
    all you really need to show is that q<1.

    remember:
    the sum of that series = (1 - q^n)/(1-q)
    when n goes to infinity then the sum = 1/(1-q)
     
  4. Mar 15, 2006 #3
    greytomato, no. Your "q" must be constant. Using your logic, the harmonic series would be convergent...
     
  5. Mar 16, 2006 #4

    MathematicalPhysicist

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    so muzza, what approach should i take here?
     
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