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Homework Help: Conditional convergence

  1. Nov 2, 2005 #1
    Hi, We're debating the question "Can a series of nonnegative numbers converge conditionally?"
    I say no becuase if all of the terms are nonnegative then they are the same as their absolute values. My classmate disagrees and says that there is a series that has nonegative terms whose absolute value diverges. I'm really confused. He won't tell us what this divergent series is and I can't come up with a counterexample of my own. I keep staring at the definition of absolute convergence and getting more confused.
    help me please.
  2. jcsd
  3. Nov 2, 2005 #2
    You are correct, your classmate is wrong.

    Suppose a_n >= 0 and that [itex]\sum_{k = 1}^{\infty} a_k[/itex] converges, and [itex]\sum_{k = 1}^{\infty} |a_k|[/itex] diverges. But |a_n| = a_n, so that [itex]\sum_{k = 1}^{\infty} |a_k| = \sum_{k = 1}^{\infty} a_k[/itex], so that that series both converges and diverges. Clearly nonsense.

    Of course there is such a series (consider [itex]\sum_{k = 1}^{\infty} |k|[/itex]), but that's totally irrelevant.
    Last edited: Nov 2, 2005
  4. Nov 2, 2005 #3
    Thanks for clearing that up. That's exactly what I thought!
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