Convergence of a Challenging Series with an Additional Factor

[simba924]

Homework Statement

\sum from n=2 to \infty of n/((n²-5)*(ln n)²)

Homework Equations

The Attempt at a Solution

I've tried Limit Comparison but I always get a limit of 0 which will not work. Ratio test doesn't help. I don't think a direct comparison can be made but that seems to be the only other option...

 

[yyat]

Hi simba924!

To simplify the problem a bit, show that

\frac{n}{n^2-5}\le\frac{2}{n}

for large n. Then apply the http://en.wikipedia.org/wiki/Integral_test_for_convergence" .

 

[simba924]

Ahh thanks. Can you explain that in a little more detail though, I don't really get it.

 

[simba924]

Hey I still need some help with this one. I'm pretty sure that \sum 2/n is not convergent because it is in form integer/n_p where p=1

Can anyone help?

 

[yyat]

Hey I still need some help with this one. I'm pretty sure that \sum 2/n is not convergent because it is in form integer/n_p where p=1

Can anyone help?

Remember, you still have the additional factor 1/(ln n)^2 (without which it would diverge). To do the integral test you have to check that

\int_2^{\infty}\frac{1}{x(\log(x))^2}

exists (you can find an explicit antiderivative, Hint: substitution).