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Math Help - Does the serie converge or not?

  1. #1
    MHF Contributor Siron's Avatar
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    Does the serie converge or not?

    Hi,

    I'm not able to check if the following serie converges or not:
    \sum_{n} \frac{1}{\sqrt{n(1+n^2)}}

    The ratio test didn't work, and I think I need a majorant or a minorant but I can't find one.
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    MHF Contributor Reckoner's Avatar
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    Re: Does the serie converge or not?

    Quote Originally Posted by Siron View Post
    Hi,

    I'm not able to check if the following serie converges or not:
    \sum_{n} \frac{1}{\sqrt{n(1+n^2)}}
    Use limit comparison with the convergent p-series \sum_n\frac1{n^{3/2}}.

    Spoiler:
    \lim_{n\to\infty}\frac{1/\sqrt{n+n^3}}{1/n^{3/2}}

    =\lim_{n\to\infty}\frac{n^{3/2}}{\sqrt{n+n^3}}

    =\lim_{n\to\infty}\sqrt{\frac{n^3}{n+n^3}}

    =\lim_{n\to\infty}\sqrt{\frac1{\frac1{n^2}+1}}
    Thanks from Siron
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    MHF Contributor Siron's Avatar
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    Re: Does the serie converge or not?

    Thanks!
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