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Could you check if these two are equivalent?

  1. Dec 15, 2011 #1
    Hi guys,

    This is from so called ratio test, which says

    The series Ʃa(n) diverges if abs{a(n+1)/a(n)} ≥ 1 for n≥N where N is some fixed integer.

    I was wondering if I could replace the condition with lim inf abs{a(n+1)/a(n)} ≥ 1.

    So basically what I'm asking is whether these two below are equivalent or not.
    1. abs{a(n+1)/a(n)} ≥ 1 for n≥N where N is some fixed integer
    2. lim inf abs{a(n+1)/a(n)} ≥ 1

    *a(n) means nth term in the sequence, and abs means absolute value.

    Thank you in advance.
  2. jcsd
  3. Dec 15, 2011 #2


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    Science Advisor

    No, it doesn't say that. You need to replace "[itex]\ge[/itex]" with ">".

    Yes, if you replace that [itex]\ge[/itex] with ">" in each.

  4. Dec 16, 2011 #3
    If it equals one, the ratio test is inconclusive.
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