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Valid application of Weierstrass Test?

  1. Mar 4, 2008 #1
    It would seem so

    1. The problem statement, all variables and given/known data

    If
    [tex]
    \sum\limits_{n = 1}^{\inf } {|{f(n,x)}|}
    [/tex] is uniformly convergent on [a,b], then is [tex]
    \sum\limits_{n = 1}^{\inf } {{f(n,x)}}
    [/tex] uniformly convergent.


    2. Relevant equations



    3. The attempt at a solution

    I said yes. And just applied the Weierstrass Test with |f(n,x)| <= |f(n,x)| (a basic comparison test)

    Should still be valid right? Since the absolutely value is Uniformly Cauchy
     
  2. jcsd
  3. Mar 4, 2008 #2

    quasar987

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    I don't follow your argument.

    Just because a series converge uniformly does not mean it satisfy Weierstrass' M test
     
  4. Mar 5, 2008 #3
    Well, forget the Weierstrass test then...
    If one is uniformly Cauchy, wouldn't that make the toher essentially uniformly Cauchy as well?
     
  5. Mar 5, 2008 #4

    quasar987

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    This sounds right! (triangle inequality)
     
  6. Dec 12, 2011 #5
    How does uniformly Cauchy apply to absolute uniform convergence
     
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