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megaray26

  • 3 years ago

Does this series converge or diverge?

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  1. megaray26
    • 3 years ago
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    \[\sum_{n = 1}^{\infty} \frac{ k \ln k }{ (k + 1)^3 }\]

  2. amistre64
    • 3 years ago
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    there is no n in it, si its just a big ole constant

  3. megaray26
    • 3 years ago
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    oops i meant k = 1

  4. amistre64
    • 3 years ago
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    :) my gut says taht the bottom is always bigger than the top and I would want to say it converges eventually. but I got no tests in mind to prove that

  5. amistre64
    • 3 years ago
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    have you tried a ratio test?

  6. megaray26
    • 3 years ago
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    Well i have, but it doesn't look like it will lead to an easier solution. It just seemed to make the problem harder, unless there is some trick that I am missing while doing so.

  7. amistre64
    • 3 years ago
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    http://www.wolframalpha.com/input/?i=sum+k*ln%28k%29%2F%28k%2B1%29%5E3%2C+k%3D1..inf says a comparison test has it converging

  8. amistre64
    • 3 years ago
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    think we would compare it to 1/k^3 ? or 1/(k+1)^3?

  9. megaray26
    • 3 years ago
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    What i had thought of using was comparing to k / k^3, or 1/ k^2, and maybe try a limit comparison test, but that didn't exactly work out. Also, for those, I'm not exactly sure if the series is less than 1 / k^3, which is what I need for convergence, but I may be wrong.

  10. amistre64
    • 3 years ago
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    a calculator would determine the bigger or less than stuff maybe \[\frac{1}{(k+1)^3}-\frac{ k \ln k }{ (k + 1)^3 }>0\] \[1-k \ln k>0\] \[1>k \ln k\]

  11. amistre64
    • 3 years ago
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    if we do a k up top we might be able to factor out a k first, but most of this is more art than science to me

  12. amistre64
    • 3 years ago
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    \[\frac{k}{(k+1)^3}-\frac{ k \ln k }{ (k + 1)^3 }>0\] \[k-k \ln k>0\] \[k(1- \ln k)>0\] might be able to work out something with that

  13. amistre64
    • 3 years ago
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    when k=1, 1(1-0)>0 when k=e, e(1-1)=0 http://www.wolframalpha.com/input/?i=y%3D1%2F%28x%2B1%29%5E3%2C+y%3Dx*ln%28x%29%2F%28x%2B1%29%5E3 hmm, the comparison seems to be smaller in both cases than the function we are looking at

  14. exraven
    • 3 years ago
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    the limit comparison test works, try\[1/k^{\frac{3}{2}}\]you will get\[\lim_{k \rightarrow \infty} \frac{k^{\frac{5}{2}}\ln k}{(k + 1)^3} = 0\]therefore the series converges

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