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Math Help - Convergence of a series of functions

  1. #1
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    Convergence of a series of functions

    Hi,

    Show that \sum \frac{x^n}{1+x^n} converges for x in [0,1).

    I'm not really sure how to get this one started.

    Thanks
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  2. #2
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    Is this true?
    x \in [0,1)\quad  \Rightarrow \quad \frac{x}{{1 + x}} \le x
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  3. #3
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    Quote Originally Posted by Plato View Post
    Is this true?
    x \in [0,1)\quad  \Rightarrow \quad \frac{x}{{1 + x}} \le x
    Yes, it is true. I think that implies

    \frac{x^n}{1+x^n} \le x^n

    Then I guess you just prove  \sum x^n converges for x in [0,1).

    So B = lim sup |an|^(1/n) = lim sup |1|^1/n = 1. So radius of convergence R = 1/1 = 1. Thus converges for (-1,1). Since [0,1) is a subset of (-1,1), I should converge in that interval.

    Is this valid?
    Last edited by tbyou87; November 3rd 2007 at 06:44 PM.
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by tbyou87 View Post
    Yes, it is true. I think that implies

    \frac{x^n}{1+x^n} \le x^n

    Then I guess you just prove  \sum x^n converges for x in [0,1).

    So B = lim sup |an|^(1/n) = lim sup |1|^1/n = 1. So radius of convergence R = 1/1 = 1. Thus converges for (-1,1). Since [0,1) is a subset of (-1,1), I should converge in that interval.

    Is this valid?
    you are working too hard, note that \sum x^n is a geometric series. what are the restraints on x for convergence?
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  5. #5
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    o yeah x element (-1,1).
    Thanks forgot about that lol.
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