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Quotient criteria and the harmonic series

  1. Jul 3, 2009 #1
    Ok so you can't apply the quotient criteria to the harmonic series because:

    [tex]lim_{k\to \infty}|\dfrac{a_{k+1}}{a_k}|[/tex]

    applied to the harmonic series:

    [tex]lim_{k\to \infty}|\dfrac{1/(k+1)}{1/k}| = lim_{k\to \infty}|\dfrac{k}{k+1}| < 1[/tex]
    which does fullfill the quotient criteria, yet the harmonic series diverges...

    So where else does it not work?
  2. jcsd
  3. Jul 3, 2009 #2


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    ??? No!
    [tex]\lim_{k\to\infty}|\frac{k}{k+1}|= 1[/tex]!

    It does NOT "fulfill the quotient criteria".

  4. Jul 3, 2009 #3
    ok, so its the fact that it converges to 1 which makes it not work?
  5. Jul 3, 2009 #4


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    What exactly do you think the ratio test says?

    It looks to me like it does exactly what it claims to do!
  6. Jul 6, 2009 #5

    Gib Z

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    The fact that the ratio test gives 1 is an inconclusive result - it does not tell us it converges or diverges. More is needed to show this series diverges.
  7. Jul 6, 2009 #6
    If you are looking to establish the divergence of the harmonic series try using the integral test.
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