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Math Help - Series convergence question

  1. #1
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    Series convergence question

    This is a very simple question, but I just wanted to to make sure. The question asks to find whether or not the series [k(k+2)]/[(k+3)^3], k=1 to infinity, converges or diverges, and if it converges, to find the sum. So I used the limit method and found that the limit as k goes to inifinity is 1. The book says that the limit has to go to 0 for it to converge, so does this mean that the series diverges since the limit is 1 (if that is in fact the answer)?
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  2. #2
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    Quote Originally Posted by virtuoso735 View Post
    This is a very simple question, but I just wanted to to make sure. The question asks to find whether or not the series [k(k+2)]/[(k+3)^3], k=1 to infinity, converges or diverges, and if it converges, to find the sum. So I used the limit method and found that the limit as k goes to inifinity is 1. The book says that the limit has to go to 0 for it to converge, so does this mean that the series diverges since the limit is 1 (if that is in fact the answer)?
    \sum{\frac{k(k+2)}{(k+3)^3}}

    note that the degree of the numerator is 2, while the degree of the denominator is 3 ... the kth term goes to 0 as k approaches infinity, but the series diverges ... do a limit comparison with the known divergent series \sum{\frac{1}{k}}
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  3. #3
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    Sorry, I posted the question wrong. The denominator should be (k+3)^2, not (k+3)^3, but I guess the same thing applies.
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  4. #4
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    much easier then ...

    \sum{\frac{k(k+2)}{(k+3)^2}}

    as k approaches infinity, kth term does not go to 0 ... series diverges.
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