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Math Help - How do i determine whether a series is convergent or divergent?

  1. #1
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    How do i determine whether a series is convergent or divergent?

    What's the systematic step that I can follow whenever they throw questions about series?
    eg. sum from n=1 to infinity n(n+2) / (n+3)^2.
    Should I find the nth term first? then calculate the limits? how?
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    The divergence test is one of the simplest methods to decide whether a series does converge or not.

    In this case, it's quite simple: compute the limit of the general term, and you'll see that is not zero, hence, the series diverges.

    Now, another proof could be bounding below the general term as follows <br />
\frac{n^{2}+2n}{n^{2}+6n+9}>\frac{2n}{16n^{2}}=\fr  ac{1}{8n}, so the series diverges (again) by direct comparison test with the Harmonic series.
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