1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?