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Math Help - Converence Sequence

  1. #1
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    Converence Sequence

    The series (1x2)/(3^2x4^2) + (3x4)/(5^2x6^2) + (5x6)/(7^2+8^2) +... is convergent?
    a) True
    b) False

    Is the answer true?

    Thanks!
    Bruce
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  2. #2
    Super Member
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    Re: Converence Sequence

    So the series is \sum_{n=1}^\infty \frac{n(n+1)}{(n+2)^2(n+3)^2}.

    If you know the limit comparison test, you should be able to apply it directly.

    With a little effort, you can use the comparison test:

    (n+2)^2 = n^2+4x+4 > n^2+4x = n(n+4)

    (n+3)^2 = n^2+6x+9 > n^2+6x+5 = (n+1)(n+5)

    so

    \frac{n(n+1)}{(n+2)^2(n+3)^2} < \frac{n(n+1)}{n(n+4)(n+1)(n+5)} = \frac{1}{(n+4)(n+5)}

    and \sum_{n=1}^\infty \frac{1}{(n+4)(n+5)} is a convergent series.

    - Hollywood
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