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Math Help - using integral test to determine convergence or divergence

  1. #1
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    using integral test to determine convergence or divergence

    Teacher says this converges but I keep getting divergence. What am I missing or messing up on?
    \sum_{n=2}^\infty\frac{ln~n}{n^3}

    This is what I get by using integration by parts:
    \int_2^\infty\frac{ln~x}{x^3}~dx=\frac{-ln~x}{2x^2}+\frac{1}{2}\int_2^\infty\frac{dx}{x^3}  =\frac{-ln~x}{2x^2}-\frac{1}{4x^2}\big|_2^\infty

    I come up with negative infinity plus a constant = infinity
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Possible actuary View Post
    Teacher says this converges but I keep getting divergence. What am I missing or messing up on?
    \sum_{n=2}^\infty\frac{ln~n}{n^3}

    This is what I get by using integration by parts:
    \int_2^\infty\frac{ln~x}{x^3}~dx=\frac{-ln~x}{2x^2}+\frac{1}{2}\int_2^\infty\frac{dx}{x^3}  =\frac{-ln~x}{2x^2}-\frac{1}{4x^2}\big|_2^\infty

    I come up with negative infinity plus a constant = infinity
    \lim_{x \to \infty} \left( \frac{- \ln x}{2x^2}-\frac{1}{4x^2} \right) = 0
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    Do not use the integral test,

    Note that,
    0\leq \frac{\ln n}{n^3} \leq \frac{n}{n^3} = \frac{1}{n^2}

    And \sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6} < \infty

    So it converges.
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    Quote Originally Posted by Jhevon View Post
    \lim_{x \to \infty} \left( \frac{-ln~x}{2x^2}-\frac{1}{4x^2} \right) = 0
    Thanks Just now saw that the denominator in the first fraction is growing larger than ln x and therefore goes to 0 along with the second fraction.
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    Quote Originally Posted by Possible actuary View Post
    Thanks Just now saw that the denominator in the first fraction is growing larger than ln x and therefore goes to 0 along with the second fraction.
    Yes, it is useful to remember that,
    \lim_{x\to \mbox{Me}} \frac{\ln x}{x} = 0
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  6. #6
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    Quote Originally Posted by ThePerfectHacker View Post
    Yes, it is useful to remember that,
    \lim_{x\to \mbox{Me}} \frac{\ln x}{x} = 0
    Interesting... What does the "Me" mean? Is it another joke? TPH=\infty?
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  7. #7
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by rualin View Post
    Interesting... What does the "Me" mean? Is it another joke? TPH=\infty?
    yes. he does that from time to time. there are quite a few whimsical fellows on this forum... heck, i'm answering one right now, Mr. I-Own-A-Smurf
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  8. #8
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Jhevon View Post
    yes. he does that from time to time. there are quite a few whimsical fellows on this forum... heck, i'm answering one right now, Mr. I-Own-A-Smurf
    I had a Smurf once. Since its skin was always blue I thought it was cold so I kept wrapping it in blankets. It died of heat prostration.

    -Dan
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  9. #9
    Newbie servantes135's Avatar
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    lol smurfs. Hey Smurf! lets go smurf those smurfity smurf smurfs and then smurf the smurfs right out!

    aint Smurfing fun!
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  10. #10
    Senior Member DivideBy0's Avatar
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    Omg it's a smurf! What should I do? Stay or run?
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