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Math Help - Function series which converges uniformly but not absolutely

  1. #1
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    Function series which converges uniformly but not absolutely

    Hello,

    I'm new here and would like some help finding a function series qualifying the conditions in the title.

    Thanks!
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  2. #2
    Super Member girdav's Avatar
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    Take f_n\left(x\right) =\frac{\left(-1\right)^n}n for x, for example, in \left[0,1\right]. The function is constant so the convergence is uniform and it doesn't converge absolutely.
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  3. #3
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    Thanks, but when I say series I mean
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Mosho View Post
    Thanks, but when I say series I mean
    That's what he gave you. Let g(x)=\sum_{n=1}^{\infty}\frac{(-1)^n}{n}. This converges uniformly to \ln\left(\frac{1}{2}\right) for a number of reasons...the easies to just state is Abel's Uniform Convergence Test -- from Wolfram MathWorld with a_n=\frac{(-1)^n}{n},f_n(x)=1.
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  5. #5
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    My bad then. I had an exam today and what you guys said is what I wrote, but most people said I was wrong.

    Thanks alot
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  6. #6
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    I have another question:

    Is this statement true/false?

    "The sum of a absolutely convergent series and a semi convergent series is a semi convergent series"

    Thanks again!
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  7. #7
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Mosho View Post
    I have another question:

    Is this statement true/false?

    "The sum of a absolutely convergent series and a semi convergent series is a semi convergent series"

    Thanks again!
    What does semi-convergent mean?
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  8. #8
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    Sorry, conditionally convergent.
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