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Limit question

  1. Oct 21, 2009 #1
    1. The problem statement, all variables and given/known data

    this is more of a question I had within a question... but here it is:

    Suppose [tex]s_{k} = s_{2k-1} + s_{2k}[/tex]

    is true and I know for a fact that s_2k has no limit.

    Would that imply that s_k has no limit as well? Or is that not enough?

    Thanks in advance.

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Oct 21, 2009 #2


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    Not enough. Take s_k=(-1)^k.
  4. Oct 21, 2009 #3
    Actually, you've provided "too much"! See, the statement "If [tex]s_{2k}[/tex] has no limit, then neither does [tex]s_k[/tex]" is the contrapositive of "If [tex]s_k[/tex] has a limit, then so does [tex]s_{2k}[/tex]," the truth of which follows quite readily for all sequences from the definition of a sequence limit.

    But this doesn't satisfy the condition that [tex]s_{2k}[/tex] not converge.
  5. Oct 21, 2009 #4


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    Good point. You are right. If s_2k doesn't converge, s_k can't converge.
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