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Math Help - Proof series diverges

  1. #1
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    Proof series diverges

    \sum_{n=1}^{\propto } \frac{3\sqrt{n}}{n}

    How to prove this series diverges?

    Many thanks!
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Re: Proof series diverges

    Quote Originally Posted by BabyMilo View Post
    \sum_{n=1}^{\propto } \frac{3\sqrt{n}}{n}

    How to prove this series diverges?

    Many thanks!
    \sum_{n=1}^\infty\frac{3\sqrt{n}}{n}= \sum_{n=1}^\infty\frac{3}{\sqrt{n}}

    \sum_{n=1}^\infty\frac{1}{\sqrt{n}} is a p-series with p=\frac{1}{2}. So, \sum_{n=1}^\infty\frac{1}{\sqrt{n}} diverges and consequently, \sum_{n=1}^\infty\frac{3}{\sqrt{n}} diverges.
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  3. #3
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    Re: Proof series diverges

    Quote Originally Posted by alexmahone View Post
    \sum_{n=1}^\infty\frac{3\sqrt{n}}{n}= \sum_{n=1}^\infty\frac{3}{\sqrt{n}}

    \sum_{n=1}^\infty\frac{1}{\sqrt{n}} is a p-series with p=\frac{1}{2}. So, \sum_{n=1}^\infty\frac{1}{\sqrt{n}} diverges and consequently, \sum_{n=1}^\infty\frac{3}{\sqrt{n}} diverges.
    thank you for your answer.
    but when i tried to do a ratio test, the answer is smaller than 1 which suggests it converges, but we know this series diverges.
    can you show me why?

    many thanks.
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  4. #4
    MHF Contributor alexmahone's Avatar
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    Re: Proof series diverges

    Quote Originally Posted by BabyMilo View Post
    thank you for your answer.
    but when i tried to do a ratio test, the answer is smaller than 1 which suggests it converges, but we know this series diverges.
    can you show me why?

    many thanks.
    Actually L = 1, which makes the ratio test inconclusive.
    Last edited by alexmahone; January 7th 2012 at 04:25 PM.
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  5. #5
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    Re: Proof series diverges

    Quote Originally Posted by alexmahone View Post
    Actually L = 1, which makes the ratio test inconclusive.
    is there another way of proofing?
    other than the p-series and the ratio test?
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  6. #6
    MHF Contributor alexmahone's Avatar
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    Re: Proof series diverges

    Quote Originally Posted by BabyMilo View Post
    is there another way of proofing?
    other than the p-series and the ratio test?
    If you know that the harmonic series diverges, you can use the comparison test.
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  7. #7
    MHF Contributor chisigma's Avatar
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    Re: Proof series diverges

    A 'direct' proof that the series \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} diverges is very comfortable. If we consider the partial sum of the first k terms we note that...

    S_{k}=\sum_{n=1}^{k} \frac{1}{\sqrt{n}} \ge k\ \frac{1}{\sqrt{k}}= \sqrt{k} (1)

    ... so that \lim_{k \rightarrow \infty} S_{k}= \infty...

    Kind regards

    \chi \sigma
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  8. #8
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Proof series diverges

    \frac{1}{\sqrt{n}} is non-negative monotone decreasing function.

    and:

    \int\frac{1}{\sqrt{n}}dn=2\sqrt{n}+C
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