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Thread: Convergence (series)

  1. #1
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    Convergence (series)

    I have to assess if $\displaystyle \sum_{n=1}^{ \infty } \frac{1}{1+ n^{2} \cdot x^{2} }$ is convergent (uniformly convergent)

    please help
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  2. #2
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    Re: Convergence (series)

    Quote Originally Posted by fqqs View Post
    I have to assess if $\displaystyle \sum_{n=1}^{ \infty } \frac{1}{1+ n^{2} \cdot x^{2} }$ is convergent (uniformly convergent)
    Think of the limit comparison test.

    What can you say about $\displaystyle \left( {\frac{{{n^2}}}{{1 + {n^2}{x^2}}}} \right) \to~? $
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    Re: Convergence (series)

    Quote Originally Posted by Plato View Post
    Think of the limit comparison test.

    What can you say about $\displaystyle \left( {\frac{{{n^2}}}{{1 + {n^2}{x^2}}}} \right) \to~? $
    It goes to $\displaystyle \frac{1}{x^{2}}$?
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    Re: Convergence (series)

    Quote Originally Posted by fqqs View Post
    It goes to $\displaystyle \frac{1}{x^{2}}$?
    What does that tell you about the values of $\displaystyle x$ for which it converges or diverges?
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  5. #5
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    Re: Convergence (series)

    it converges for every $\displaystyle x \ne 0 $


    ?
    Last edited by fqqs; May 26th 2012 at 11:14 AM.
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  6. #6
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    Re: Convergence (series)

    Quote Originally Posted by fqqs View Post
    it converges for every $\displaystyle x \ne 0 $
    Correct
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  7. #7
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    Re: Convergence (series)

    but that only means that it is pointwisely convergent for every $\displaystyle x \ne 0$, yes?

    so i need to check if it is pointwisely convergent somewhere , yes?

    maybe I should use cases when $\displaystyle |x| < 1$ and when $\displaystyle |x| \ge 1 $ and use Weierstrass?
    Last edited by fqqs; May 26th 2012 at 12:22 PM.
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  8. #8
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    Re: Convergence (series)

    upppp
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