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Continuity of a power-series function

  1. Jan 25, 2013 #1
    1. The problem statement, all variables and given/known data

    Prove the function:
    [itex]g(x)=\sum_{n=1}^{\infty }\frac{1}{^{n^{0.5}}}(x^{2n}-x^{2n+1})[/itex]

    is continuous in [0,1].


    2. The attempt at a solution

    I tried to look at this functions as:

    [itex]g(x)=(1-x)\sum_{n=1}^{\infty }\frac{1}{^{n^{0.5}}}x^{2n}[/itex]

    but I couldn't find a way solving it from here.
    Finding the radius of convergence (which is 1) didn't help a lot...
     
  2. jcsd
  3. Jan 25, 2013 #2

    jbunniii

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    Science Advisor
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    Gold Member

    Yes, writing it as
    $$g(x) = (1-x) \sum_{n=1}^\infty \frac{1}{n^{0.5}} x^{2n}$$
    is a good start. Let us introduce the notation
    $$p(x) = \sum_{n=1}^\infty \frac{1}{n^{0.5}} x^{2n}$$
    This is a power series. If its radius of convergence is 1, do you know a theorem which tells you that p is continuous on (-1,1)? If so, then you only need to worry about the point x = 1. Does the series converge at x = 1?
     
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