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Absolute convergence of series?

  1. Nov 12, 2012 #1
    The question is: Show that if [itex]\sum[/itex]an from n=1 to ∞ converges absolutely, then [itex]\sum[/itex]an2 from n=1 to converges absolutely.

    I'm not sure which approach to take with this.

    I am thinking that since Ʃan converges absolutely, |an| can be either -an or an and for Ʃan2, an can be either negative or positive. But i'm not sure where to go with this.

    Any help is appreciated.

    Thank you.
     
  2. jcsd
  3. Nov 12, 2012 #2

    micromass

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    You know that if [itex]a_n[/itex] is small (=smaller than 1), then [itex]a_n^2[/itex] is even smaller. So the squared series is smaller than the original series.

    Can you do something with this idea??
     
  4. Nov 13, 2012 #3
    So you mean a_n^2 would be bounded from above for smaller values? But what if it's greater than 1?
     
  5. Nov 13, 2012 #4

    micromass

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    Yes, if it's greater than 1 then there is a problem. Try to find a way to solve this.
     
  6. Nov 13, 2012 #5
    Do you think I can use what I stated in my first post?
     
  7. Nov 13, 2012 #6

    micromass

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    Uh, not really. Well,it's not wrong what you said, but it won't help you much further.
     
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