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Searching for a particular kind of convergent sequence

  1. Dec 24, 2009 #1
    I want an example of a complex sequence [tex](x_n)[/tex] which converges to 0 but is not in [tex]^p[/tex], for [tex]p\ge 1[/tex] i.e. the series [tex]\sum |x_n|^p[/tex] is never convergent for any [tex]p\ge 1[/tex]. Can someone provide an example please?
  2. jcsd
  3. Dec 24, 2009 #2
    I suspect 1/log(n+1) will work, but I haven't checked divergence for p > 1.

    *EDIT* I'm fairly certain it works. I'll let you figure out the estimates needed to demonstrate divergence (hint: you don't need obscure series tests).
    Last edited: Dec 24, 2009
  4. Dec 24, 2009 #3
    I am having trouble establishing the divergence. Can you be more explicit? Thanks.
  5. Dec 24, 2009 #4


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    For divergence I would guess (1/log(n))p > 1/n for sufficiently large n.
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