Here's the question you clicked on:
megaray26
Does this series converge or diverge?
\[\sum_{n = 1}^{\infty} \frac{ k \ln k }{ (k + 1)^3 }\]
there is no n in it, si its just a big ole constant
:) 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
have you tried a ratio test?
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.
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
think we would compare it to 1/k^3 ? or 1/(k+1)^3?
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.
a calculator would determine the bigger or less than stuff maybe \[\frac{1}{(k+1)^3}-\frac{ k \ln k }{ (k + 1)^3 }>0\] \[1-k \ln k>0\] \[1>k \ln k\]
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
\[\frac{k}{(k+1)^3}-\frac{ k \ln k }{ (k + 1)^3 }>0\] \[k-k \ln k>0\] \[k(1- \ln k)>0\] might be able to work out something with that
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
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