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Convergence of a very simple series

  1. Oct 23, 2005 #1
    Hi, there's a really simple looking series that I don't know how to deal with.
    [tex]
    \sum\limits_{n = 2}^\infty {\frac{{n^2 }}{{1 - n^3 }}}
    [/tex]
    How would I determine whether or not this series converges by using some standard convergence tests? If a_n = (n^2)/(1-n^3) then the numerator is always positive while the denominator is always negative so that a_n is always negative. So I can't think of a way to use the comparison test, limit comparison test etc. Since a_n looks like 1/n I have a feeling that the series diverges. But I can't think of any tests to use to verify whether or not my hypothesis is correct. At first I thought about using the absolute convergence test but if a series is not absolutely convergent, it can still be convergent so that didn't really help.
    Can someone help me with this one?

    Note: I would like to be able to do this without the integral test if possible.
     
    Last edited: Oct 23, 2005
  2. jcsd
  3. Oct 23, 2005 #2

    arildno

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    Rewrite your series S as: [itex]S=-s, s=\sum_{n=2}^{\infty}\frac{n^{2}}{n^{3}-1}[/tex]
    S diverges or converges with s.
     
  4. Oct 23, 2005 #3
    Oh ok, thanks for the help arildno.
     
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