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Convergence of a functional series (analysis)

  1. Dec 16, 2012 #1
    1. The problem statement, all variables and given/known data

    Determine whether the following functional series is pointwise and/or uniformly convergent:

    [itex]\sum_{n=1}^\infty \frac{x}{n} (x\in\mathbb{R})[/itex]

    2. Relevant equations

    3. The attempt at a solution

    My answer to this seems very straightforward and I would be very grateful if someone could let me know if the method is correct. We can re-write the series as:

    [itex]x \sum_{n=1}^\infty \frac{1}{n}[/itex]

    We know the harmonic series (of numbers) [itex]\sum_{n=1}^\infty \frac{1}{n}[/itex] is divergent, so therefore the series cannot be pointwise or uniformly convergent.

    Is this correct? Thanks!
     
  2. jcsd
  3. Dec 16, 2012 #2

    sharks

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    Gold Member

    You could also use the p-series test to arrive at the same conclusion.
     
  4. Dec 16, 2012 #3
    Thanks. I just noticed something: if x=0, then the series is just zero, so it converges. Given that the series isn't convergent for *all* values of x in the domain, is it still correct to say the functional series is divergent? I suppose it's a question of terminology. Thanks for the help.
     
  5. Dec 16, 2012 #4

    pasmith

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    Homework Helper

    It's pointwise convergent at points where it converges, and pointwise divergent where it doesn't converge.
     
  6. Dec 16, 2012 #5

    sharks

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    Gold Member

    Based on your question, i would say that the correct answer is: point-wise convergent at x = 0. (which would imply that at any other values of x, the series is divergent)
     
  7. Dec 17, 2012 #6
    Ok, thanks!
     
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