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Math Help - Determining convergence or divergence

  1. #1
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    Determining convergence or divergence

    The problem asks whether this sum will converge or diverge:

    \sum_{n=1}^{\infty} \frac {n(n+1)} {\sqrt {n^3+2n^2)}}

    What I had done from this point was to attempt to simplify the sum, by factoring the denominator so that instead of having:

    \sqrt {n^3+2n^2)}

    I had:

    \sqrt {n^2(n+2)} which becomes: n\sqrt {(n+2)}

    So then back to the initial fraction, the factor of n would cancel and what would be left is:

    \frac {(n+1)} {\sqrt {n+2)}}

    Is it possible to simply take the limit of this or just even say that because the denominator is raised to a power of \frac {1} {2} that the series must diverge? I am tempted to simply just note the power value and say that the sum must diverge then.

    Thanks for the help, it's much appreciated.
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    it diverges, we can bound the general term as follows: \frac{n^{2}+n}{\sqrt{n^{3}+2n^{2}}}>\frac{n^{2}}{\  sqrt{2n^{3}+2n^{3}}}=\frac{\sqrt{n}}{2},
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  3. #3
    Member Miss's Avatar
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    The limit comparison test with \sum \frac{1}{n} also works.
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