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Math Help - Is the following series absolutely convergent or not

  1. #1
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    Is the following series absolutely convergent or not

    Hi,

    The series is : \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2n+1}

    The absolut series is : \sum_{n=1}^{\infty} \frac{1}{2n+1}

    But I'm stuck at this very (easy ?) step : s_{n} = \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + ...

    But I can't determine, in my head, if it is convergent or not. I feel really dumb.

    If I compare to the harmonic series, each term is lower.

    The "quotient test" gives me : \frac{2n+1}{2n+3} \textless 1

    What is the tiny thing I missed ?

    Thanks for your help !
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  2. #2
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    Re: Is the following series absolutely convergent or not

    the series is not absolutely convergent ... use the integral test.
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  3. #3
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    Re: Is the following series absolutely convergent or not

    Quote Originally Posted by NZAU1984 View Post
    The series is : \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2n+1}
    The absolut series is : \sum_{n=1}^{\infty} \frac{1}{2n+1}
    Note that \sum_{n=1}^{\infty} \frac{1}{2n+1}>\frac{1}{4}\sum_{n=1}^{\infty} \frac{1}{n}
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