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Convergence/Divergence and Reordering of an Alternating Series

  1. Oct 24, 2011 #1
    I was just thinking about the following series:


    I'm not familiar with any other series like this one(other than the alternating harmonic), and I was curious as to whether or not it would be convergent, and if reordering it to


    would change its convergence.

    If its divergent, which it seems it would be, is it divergent to negative or positive infinity?

    This is probably a basic question, but I'm unable to figure it out right now. Any takers?
    Last edited: Oct 24, 2011
  2. jcsd
  3. Oct 24, 2011 #2


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    It is divergent to no particular value. It can be made to approach anything by suitable rearrangement. Examples:
    -1+2 = 1, -3+4=1, etc. -> +∞.
    0-1=-1, 2-3=-1, 4-5=-1, etc. -> -∞.
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