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Convergence tests for sequences not series

  1. Jan 6, 2005 #1


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    I'm trying to find out tests with regards to determining if a limit of a sequence exists or not (ie convergence of sequences), since evaluating a particular limit may not always possible.

    For example it seems to me that if for a particular sequence a, if
    limn->infty a(n+1)/a(n) = 1, then limn->infty a(n) exists.

    It also seems like if
    limn->infty a(n+1)/a(n) >1, then limn->infty a(n) = infty.

    This make sense to me, but I've been searching online for theorems such as these to no avail. Everything I see is with regards to the convergence of series, but not sequences.

    Can someone point me to the relevant theorems? Thanks!
  2. jcsd
  3. Jan 6, 2005 #2
    Nope, let a_n = n.
  4. Jan 6, 2005 #3
  5. Jan 6, 2005 #4


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    Thanks for the correction Muzza. Thanks Tenaliraman. I've found what I was looking for.
  6. Feb 12, 2005 #5


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    how about asking for a bounded counterexample to your conjecture.
  7. Mar 21, 2011 #6
    no need of any theorems ,the definition of a convergent sequence is lim n->infnty x=l
    l=limit of the sequence,just find the limit,if it exists,if it is unique,then te sequence is convergent..............
  8. Mar 21, 2011 #7
    For the first one, consider

    The sequence defined by a(n) = n+1*10^(-n).

    Lim a(n+1)/a(n) = 1(I hope I didn't screw that up), but clearly the sequence is unbounded.

    Not sure about the second.

    But, if I have limit laws correct(not sure if you can treat two terms of the same series like this..).

    lim a(n+ 1)/a(n) = lim a(n+1)/lim a(n) > 1, so lim a(n+1) > lim a(n),

    I'm not sure what that means.
    Last edited: Mar 21, 2011
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