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Homework Help: Proof of convergence

  1. Feb 16, 2008 #1
    1. The problem statement, all variables and given/known data
    Prove that if a_n>=0 and summation a_n converges, then summation (a_n)^2 also converges.

    3. The attempt at a solution
    (Note: When I say "lim" please assume the limit as n-->infinity). I just want it to be a little clearer to read)

    If summation a_n converges, then lim(a_n)=0. If lim(a_n)=0, then lim(a_n)^2=0.
    If summation (a_n)^2 diverges, then lim(a_n)^2 does not equal 0. But lim(a_n)^2=0, so summation (a_n)^2 must converge.

    Can anyone let me know if this is a valid proof? I'm not sure how else to prove it otherwise.....thank you.
    Last edited: Feb 16, 2008
  2. jcsd
  3. Feb 16, 2008 #2
    I'm having some trouble following those lines. What about [tex]a_{n}=\frac{1}{\sqrt{n}}[/tex]?
    Last edited: Feb 16, 2008
  4. Feb 16, 2008 #3
    Oh, that does go against my proof. I don't know how else to prove it then. Any suggestions?

    (I also edited a bit of my original post so that it would be a bit easier to read, hopefully)
  5. Feb 16, 2008 #4
  6. Feb 16, 2008 #5


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    This is definitely NOT true!

    However, it is true that if [itex]\sum a_n[/itex] converges then [itex]lim a_n= 0[/itex].
    For sufficiently large n, an< 1 and so an2< |an|.
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