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megaray26
 2 years ago
Does this series converge or diverge?
megaray26
 2 years ago
Does this series converge or diverge?

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megaray26
 2 years ago
Best ResponseYou've already chosen the best response.0\[\sum_{n = 1}^{\infty} \frac{ k \ln k }{ (k + 1)^3 }\]

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1there is no n in it, si its just a big ole constant

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1:) 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

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1have you tried a ratio test?

megaray26
 2 years ago
Best ResponseYou've already chosen the best response.0Well 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.

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1http://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

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1think we would compare it to 1/k^3 ? or 1/(k+1)^3?

megaray26
 2 years ago
Best ResponseYou've already chosen the best response.0What 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.

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1a calculator would determine the bigger or less than stuff maybe \[\frac{1}{(k+1)^3}\frac{ k \ln k }{ (k + 1)^3 }>0\] \[1k \ln k>0\] \[1>k \ln k\]

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1if 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

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1\[\frac{k}{(k+1)^3}\frac{ k \ln k }{ (k + 1)^3 }>0\] \[kk \ln k>0\] \[k(1 \ln k)>0\] might be able to work out something with that

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.1when k=1, 1(10)>0 when k=e, e(11)=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

exraven
 2 years ago
Best ResponseYou've already chosen the best response.0the 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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